2017
2017
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Paper 1, Section I, F
2017 commentGiven an increasing sequence of non-negative real numbers , let
Prove that if as for some then also as
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Paper 1, Section II, F
2017 comment(a) Let be a non-negative and decreasing sequence of real numbers. Prove that converges if and only if converges.
(b) For , prove that converges if and only if .
(c) For any , prove that
(d) The sequence is defined by and for . For any , prove that
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Paper 1, Section I,
2017 commentShow that if the power series converges for some fixed , then it converges absolutely for every satisfying .
Define the radius of convergence of a power series.
Give an example of and an example of such that converges and diverges. [You may assume results about standard series without proof.] Use this to find the radius of convergence of the power series .
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Paper 1, Section II, D
2017 comment(a) State the Intermediate Value Theorem.
(b) Define what it means for a function to be differentiable at a point . If is differentiable everywhere on , must be continuous everywhere? Justify your answer.
State the Mean Value Theorem.
(c) Let be differentiable everywhere. Let with .
If , prove that there exists such that . [Hint: consider the function defined by
if and
If additionally , deduce that there exists such that .
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Paper 1, Section II, D
2017 commentLet with and let .
(a) Define what it means for to be continuous at .
is said to have a local minimum at if there is some such that whenever and .
If has a local minimum at and is differentiable at , show that .
(b) is said to be convex if
for every and . If is convex, and , prove that
for every .
Deduce that if is convex then is continuous.
If is convex and has a local minimum at , prove that has a global minimum at , i.e., that for every . [Hint: argue by contradiction.] Must be differentiable at ? Justify your answer.
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Paper 1, Section II, E
2017 commentLet be a bounded function defined on the closed, bounded interval of . Suppose that for every there is a dissection of such that , where and denote the lower and upper Riemann sums of for the dissection . Deduce that is Riemann integrable. [You may assume without proof that for all dissections and of
Prove that if is continuous, then is Riemann integrable.
Let be a bounded continuous function. Show that for any , the function defined by
is Riemann integrable.
Let be a differentiable function with one-sided derivatives at the endpoints. Suppose that the derivative is (bounded and) Riemann integrable. Show that
[You may use the Mean Value Theorem without proof.]
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Paper 2, Section I, C
2017 comment(a) The numbers satisfy
where are given constants. Find in terms of and .
(b) The numbers satisfy
where are given non-zero constants and are given constants. Let and , where . Calculate , and hence find in terms of and .
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Paper 2, Section I,
2017 commentConsider the function
defined for and , where is a non-zero real constant. Show that is a stationary point of for each . Compute the Hessian and its eigenvalues at .
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Paper 2, Section II, C
2017 commentThe current at time in an electrical circuit subject to an applied voltage obeys the equation
where and are the constant resistance, inductance and capacitance of the circuit with and .
(a) In the case and , show that there exist time-periodic solutions of frequency , which you should find.
(b) In the case , the Heaviside function, calculate, subject to the condition
the current for , assuming it is zero for .
(c) If and , where is as in part (a), show that there is a timeperiodic solution of period and calculate its maximum value .
(i) Calculate the energy dissipated in each period, i.e., the quantity
Show that the quantity defined by
satisfies .
(ii) Write down explicitly the general solution for all , and discuss the relevance of to the large time behaviour of .
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Paper 2, Section II,
2017 commentLet and be two solutions of the differential equation
where and are given. Show, using the Wronskian, that
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either there exist and , not both zero, such that vanishes for all ,
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or given and , there exist and such that satisfies the conditions and .
Find power series and such that an arbitrary solution of the equation
can be written as a linear combination of and .
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Paper 2, Section II, C
2017 comment(a) Solve subject to . For which is the solution finite for all ?
Let be a positive constant. By considering the lines for constant , or otherwise, show that any solution of the equation
is of the form for some function .
Solve the equation
subject to for a given function . For which is the solution bounded on ?
(b) By means of the change of variables and for appropriate real numbers , show that the equation
can be transformed into the wave equation
where is defined by . Hence write down the general solution of .
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Paper 2, Section II, C
2017 comment(a) Consider the system
for . Find the critical points, determine their type and explain, with the help of a diagram, the behaviour of solutions for large positive times .
(b) Consider the system
for . Rewrite the system in polar coordinates by setting and , and hence describe the behaviour of solutions for large positive and large negative times.
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Paper 4, Section I, A
2017 commentConsider a system of particles with masses and position vectors . Write down the definition of the position of the centre of mass of the system. Let be the total kinetic energy of the system. Show that
where is the total mass and is the position vector of particle with respect to .
The particles are connected to form a rigid body which rotates with angular speed about an axis through , where . Show that
where and is the moment of inertia of particle about .
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Paper 4, Section I, A
2017 commentA tennis ball of mass is projected vertically upwards with initial speed and reaches its highest point at time . In addition to uniform gravity, the ball experiences air resistance, which produces a frictional force of magnitude , where is the ball's speed and is a positive constant. Show by dimensional analysis that can be written in the form
for some function of a dimensionless quantity .
Use the equation of motion of the ball to find .
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Paper 4, Section II, A
2017 comment(a) A photon with energy in the laboratory frame collides with an electron of rest mass that is initially at rest in the laboratory frame. As a result of the collision the photon is deflected through an angle as measured in the laboratory frame and its energy changes to .
Derive an expression for in terms of and .
(b) A deuterium atom with rest mass and energy in the laboratory frame collides with another deuterium atom that is initially at rest in the laboratory frame. The result of this collision is a proton of rest mass and energy , and a tritium atom of rest mass . Show that, if the proton is emitted perpendicular to the incoming trajectory of the deuterium atom as measured in the laboratory frame, then
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Paper 4, Section II, A
2017 commentA particle of unit mass moves under the influence of a central force. By considering the components of the acceleration in polar coordinates prove that the magnitude of the angular momentum is conserved. [You may use . ]
Now suppose that the central force is derived from the potential , where is a constant.
(a) Show that the total energy of the particle can be written in the form
Sketch in the cases and .
(b) The particle is projected from a very large distance from the origin with speed and impact parameter . [The impact parameter is the distance of closest approach to the origin in absence of any force.]
(i) In the case , sketch the particle's trajectory and find the shortest distance between the particle and the origin, and the speed of the particle when .
(ii) In the case , sketch the particle's trajectory and find the corresponding shortest distance between the particle and the origin, and the speed of the particle when .
(iii) Find and in terms of and . [In answering part (iii) you should assume that is the same in parts (i) and (ii).]
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Paper 4, Section II, A
2017 comment(a) Consider an inertial frame , and a frame which rotates with constant angular velocity relative to . The two frames share a common origin. Identify each term in the equation
(b) A small bead of unit mass can slide without friction on a circular hoop of radius . The hoop is horizontal and rotating with constant angular speed about a fixed vertical axis through a point on its circumference.
(i) Using Cartesian axes in the rotating frame , with origin at and -axis along the diameter of the hoop through , write down the position vector of in terms of and the angle shown in the diagram .

(ii) Working again in the rotating frame, find, in terms of and , an expression for the horizontal component of the force exerted by the hoop on the bead.
(iii) For what value of is the bead in stable equilibrium? Find the frequency of small oscillations of the bead about that point.
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Paper 4, Section II, A
2017 comment(a) A rocket moves in a straight line with speed and is subject to no external forces. The rocket is composed of a body of mass and fuel of mass , which is burnt at constant rate and the exhaust is ejected with constant speed relative to the rocket. Show that
Show that the speed of the rocket when all its fuel is burnt is
where and are the speed of the rocket and the mass of the fuel at .
(b) A two-stage rocket moves in a straight line and is subject to no external forces. The rocket is initially at rest. The masses of the bodies of the two stages are and , with , and they initially carry masses and of fuel. Both stages burn fuel at a constant rate when operating and the exhaust is ejected with constant speed relative to the rocket. The first stage operates first, until all its fuel is burnt. The body of the first stage is then detached with negligible force and the second stage ignites.
Find the speed of the second stage when all its fuel is burnt. For compare it with the speed of the rocket in part (a) in the case . Comment on the case .
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Paper 3, Section I, E
2017 commentLet be distinct elements of . Write down the Möbius map that sends to , respectively. [Hint: You need to consider four cases.]
Now let be another element of distinct from . Define the cross-ratio in terms of .
Prove that there is a circle or line through and if and only if the cross-ratio is real.
[You may assume without proof that Möbius maps map circles and lines to circles and lines and also that there is a unique circle or line through any three distinct points of
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Paper 3, Section I, E
2017 commentWhat does it mean to say that is a normal subgroup of the group ? For a normal subgroup of define the quotient group . [You do not need to verify that is a group.]
State the Isomorphism Theorem.
Let
be the group of invertible upper-triangular real matrices. By considering a suitable homomorphism, show that the subset
of is a normal subgroup of and identify the quotient .
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Paper 3, Section II, E
2017 commentLet be a normal subgroup of a finite group of prime index .
By considering a suitable homomorphism, show that if is a subgroup of that is not contained in , then is a normal subgroup of of index .
Let be a conjugacy class of that is contained in . Prove that is either a conjugacy class in or is the disjoint union of conjugacy classes in .
[You may use standard theorems without proof.]
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Paper 3 , Section II, E
2017 commentState Lagrange's theorem. Show that the order of an element in a finite group is finite and divides the order of .
State Cauchy's theorem.
List all groups of order 8 up to isomorphism. Carefully justify that the groups on your list are pairwise non-isomorphic and that any group of order 8 is isomorphic to one on your list. [You may use without proof the Direct Product Theorem and the description of standard groups in terms of generators satisfying certain relations.]
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Paper 3, Section II,
2017 comment(a) Let be a finite group acting on a finite set . State the Orbit-Stabiliser theorem. [Define the terms used.] Prove that
where is the number of distinct orbits of under the action of .
Let , and for , let .
Show that
and deduce that
(b) Let be the group of rotational symmetries of the cube. Show that has 24 elements. [If your proof involves calculating stabilisers, then you must carefully verify such calculations.]
Using , find the number of distinct ways of colouring the faces of the cube red, green and blue, where two colourings are distinct if one cannot be obtained from the other by a rotation of the cube. [A colouring need not use all three colours.]
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Paper 3, Section II, E
2017 commentProve that every element of the symmetric group is a product of transpositions. [You may assume without proof that every permutation is the product of disjoint cycles.]
(a) Define the sign of a permutation in , and prove that it is well defined. Define the alternating group .
(b) Show that is generated by the set .
Given , prove that the set if and are coprime.
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Paper 4, Section I, D
2017 comment(a) Show that for all positive integers and , either or .
(b) If the positive integers satisfy , show that at least one of and must be divisible by 3 . Can both and be odd?
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Paper 4, Section I, D
2017 comment(a) Give the definitions of relation and equivalence relation on a set .
(b) Let be the set of ordered pairs where is a non-empty subset of and . Let be the relation on defined by requiring if the following two conditions hold:
(i) is finite and
(ii) there is a finite set such that for all .
Show that is an equivalence relation on .
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Paper 4, Section II, D
2017 comment(a) State and prove the Fermat-Euler Theorem. Deduce Fermat's Little Theorem. State Wilson's Theorem.
(b) Let be an odd prime. Prove that is solvable if and only if .
(c) Let be prime. If and are non-negative integers with , prove that
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Paper 4, Section II, D
2017 comment(a) Define what it means for a set to be countable.
(b) Let be an infinite subset of the set of natural numbers . Prove that there is a bijection .
(c) Let be the set of natural numbers whose decimal representation ends with exactly zeros. For example, and . By applying the result of part (b) with , construct a bijection . Deduce that the set of rationals is countable.
(d) Let be an infinite set of positive real numbers. If every sequence of distinct elements with for each has the property that
prove that is countable.
[You may assume without proof that a countable union of countable sets is countable.]
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Paper 4, Section II, 7D
2017 comment(a) For positive integers with , show that
giving an explicit formula for . [You may wish to consider the expansion of
(b) For a function and each integer , the function is defined by
For any integer let . Show that and for all and .
Show that for each integer and each ,
Deduce that for each integer ,
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Paper 4, Section II, D
2017 commentLet be a sequence of real numbers.
(a) Define what it means for to converge. Define what it means for the series to converge.
Show that if converges, then converges to 0 .
If converges to , show that
(b) Suppose for every . Let and .
Show that does not converge.
Give an example of a sequence with and for every such that converges.
If converges, show that .
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Paper 2, Section I, F
2017 commentLet be a non-negative integer-valued random variable such that .
Prove that
[You may use any standard inequality.]
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Paper 2, Section I, F
2017 commentLet and be real-valued random variables with joint density function
(i) Find the conditional probability density function of given .
(ii) Find the expectation of given .
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Paper 2, Section II, F
2017 commentFor a positive integer , and , let
(a) For fixed and , show that is a probability mass function on and that the corresponding probability distribution has mean and variance .
(b) Let . Show that, for any ,
Show that the right-hand side of is a probability mass function on .
(c) Let and let with . For all , find integers and such that
[You may use the Central Limit Theorem.]
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Paper 2, Section II, 10F
2017 comment(a) For any random variable and and , show that
For a standard normal random variable , compute and deduce that
(b) Let . For independent random variables and with distributions and , respectively, compute the probability density functions of and .
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Paper 2, Section II, F
2017 commentLet . The Curie-Weiss Model of ferromagnetism is the probability distribution defined as follows. For , define random variables with values in such that the probabilities are given by
where is the normalisation constant
(a) Show that for any .
(b) Show that . [You may use for all without proof. ]
(c) Let . Show that takes values in , and that for each the number of possible values of such that is
Find for any .
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Paper 2, Section II, 12F
2017 comment(a) Let . For , let be the first time at which a simple symmetric random walk on with initial position at time 0 hits 0 or . Show . [If you use a recursion relation, you do not need to prove that its solution is unique.]
(b) Let be a simple symmetric random walk on starting at 0 at time . For , let be the first time at which has visited distinct vertices. In particular, . Show for . [You may use without proof that, conditional on , the random variables have the distribution of a simple symmetric random walk starting at .]
(c) For , let be the circle graph consisting of vertices and edges between and where is identified with 0 . Let be a simple random walk on starting at time 0 from 0 . Thus and conditional on the random variable is with equal probability (identifying with ).
The cover time of the simple random walk on is the first time at which the random walk has visited all vertices. Show that .
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Paper 3, Section I, B
2017 commentUse the change of variables to evaluate
where is the region of the -plane bounded by the two line segments:
and the curve
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Paper 3, Section , B
2017 comment(a) The two sets of basis vectors and (where ) are related by
where are the entries of a rotation matrix. The components of a vector with respect to the two bases are given by
Derive the relationship between and .
(b) Let be a array defined in each (right-handed orthonormal) basis. Using part (a), state and prove the quotient theorem as applied to .
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Paper 3, Section II, B
2017 comment(a) The time-dependent vector field is related to the vector field by
where . Show that
(b) The vector fields and satisfy . Show that .
(c) The vector field satisfies . Show that
where
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Paper 3, Section II, B
2017 commentBy a suitable choice of in the divergence theorem
show that
for any continuously differentiable function .
For the curved surface of the cone
show that .
Verify that holds for this cone and .
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Paper 3, Section II, B
2017 comment(a) Let be a smooth curve parametrised by arc length . Explain the meaning of the terms in the equation
where is the curvature of the curve.
Now let . Show that there is a scalar (the torsion) such that
and derive an expression involving and for .
(b) Given a (nowhere zero) vector field , the field lines, or integral curves, of are the curves parallel to at each point . Show that the curvature of the field lines of satisfies
where .
(c) Use to find an expression for the curvature at the point of the field lines of .
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Paper 3, Section II, B
2017 commentLet be a piecewise smooth closed surface in which is the boundary of a volume .
(a) The smooth functions and defined on satisfy
in and on . By considering an integral of , where , show that .
(b) The smooth function defined on satisfies on , where is the function in part (a) and is constant. Show that
where is the function in part (a). When does equality hold?
(c) The smooth function satisfies
in and on for all . Show that
with equality only if in .
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Paper 1, Section I, A
2017 commentConsider with and , where .
(a) Prove algebraically that the modulus of is and that the argument is . Obtain these results geometrically using the Argand diagram.
(b) Obtain corresponding results algebraically and geometrically for .
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Paper 1, Section I, C
2017 commentLet and be real matrices.
Show that .
For any square matrix, the matrix exponential is defined by the series
Show that . [You are not required to consider issues of convergence.]
Calculate, in terms of and , the matrices and in the series for the matrix product
Hence obtain a relation between and which necessarily holds if is an orthogonal matrix.
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Paper 1, Section II, A
2017 comment(a) Define the vector product of the vectors and in . Use suffix notation to prove that
(b) The vectors are defined by , where and are fixed vectors with and , and is a positive constant.
(i) Write as a linear combination of and . Further, for , express in terms of and . Show, for , that .
(ii) Let be the point with position vector . Show that lie on a pair of straight lines.
(iii) Show that the line segment is perpendicular to . Deduce that is parallel to .
Show that as if , and give a sketch to illustrate the case .
(iv) The straight line through the points and makes an angle with the straight line through the points and . Find in terms of .
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Paper 1, Section II, B
2017 comment(a) Show that the eigenvalues of any real square matrix are the same as the eigenvalues of .
The eigenvalues of are and the eigenvalues of are , . Determine, by means of a proof or a counterexample, whether the following are necessary valid: (i) ; (ii) .
(b) The matrix is given by
where and are orthogonal real unit vectors and is the identity matrix.
(i) Show that is an eigenvector of , and write down a linearly independent eigenvector. Find the eigenvalues of and determine whether is diagonalisable.
(ii) Find the eigenvectors and eigenvalues of .
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Paper 1, Section II, B
2017 comment(a) Show that a square matrix is anti-symmetric if and only if for every vector .
(b) Let be a real anti-symmetric matrix. Show that the eigenvalues of are imaginary or zero, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal (in the sense that , where the dagger denotes the hermitian conjugate).
(c) Let be a non-zero real anti-symmetric matrix. Show that there is a real non-zero vector a such that .
Now let be a real vector orthogonal to . Show that for some real number .
The matrix is defined by the exponential series Express and in terms of and .
[You are not required to consider issues of convergence.]
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Paper 1, Section II,
2017 comment(a) Given consider the linear transformation which maps
Express as a matrix with respect to the standard basis , and determine the rank and the dimension of the kernel of for the cases (i) , where is a fixed number, and (ii) .
(b) Given that the equation
where
has a solution, show that .
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Paper 3, Section I, G
2017 commentWhat does it mean to say that a metric space is complete? Which of the following metric spaces are complete? Briefly justify your answers.
(i) with the Euclidean metric.
(ii) with the Euclidean metric.
(iii) The subset
with the metric induced from the Euclidean metric on .
Write down a metric on with respect to which is not complete, justifying your answer.
[You may assume throughout that is complete with respect to the Euclidean metric.
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Paper 2, Section I, G
2017 commentLet . What does it mean to say that a sequence of real-valued functions on is uniformly convergent?
Let be functions.
(a) Show that if each is continuous, and converges uniformly on to , then is also continuous.
(b) Suppose that, for every converges uniformly on . Need converge uniformly on ? Justify your answer.
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Paper 4, Section I, G
2017 commentState the chain rule for the composition of two differentiable functions and .
Let be differentiable. For , let . Compute the derivative of . Show that if throughout , then for some function .
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Paper 1, Section II, G
2017 commentWhat does it mean to say that a real-valued function on a metric space is uniformly continuous? Show that a continuous function on a closed interval in is uniformly continuous.
What does it mean to say that a real-valued function on a metric space is Lipschitz? Show that if a function is Lipschitz then it is uniformly continuous.
Which of the following statements concerning continuous functions are true and which are false? Justify your answers.
(i) If is bounded then is uniformly continuous.
(ii) If is differentiable and is bounded, then is uniformly continuous.
(iii) There exists a sequence of uniformly continuous functions converging pointwise to .
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Paper 2, Section II, G
2017 commentLet be a real vector space. What is a norm on ? Show that if is a norm on , then the maps for ) and (for ) are continuous with respect to the norm.
Let be a subset containing 0 . Show that there exists at most one norm on for which is the open unit ball.
Suppose that satisfies the following two properties:
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if is a nonzero vector, then the line meets in a set of the form for some ;
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if and then .
Show that there exists a norm for which is the open unit ball.
Identify in the following two cases:
(i) for all .
(ii) the interior of the square with vertices .
Let be the set
Is there a norm on for which is the open unit ball? Justify your answer.
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Paper 4, Section II, G
2017 commentLet be a nonempty open set. What does it mean to say that a function is differentiable?
Let be a function, where is open. Show that if the first partial derivatives of exist and are continuous on , then is differentiable on .
Let be the function
Determine, with proof, where is differentiable.
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Paper 3, Section II, G
2017 commentWhat is a contraction map on a metric space ? State and prove the contraction mapping theorem.
Let be a complete non-empty metric space. Show that if is a map for which some iterate is a contraction map, then has a unique fixed point. Show that itself need not be a contraction map.
Let be the function
Show that has a unique fixed point.
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Paper 4, Section I, 4F
2017 commentLet be a star-domain, and let be a continuous complex-valued function on . Suppose that for every triangle contained in we have
Show that has an antiderivative on .
If we assume instead that is a domain (not necessarily a star-domain), does this conclusion still hold? Briefly justify your answer.
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Paper 2, Section II, 13A
2017 commentState the residue theorem.
By considering
with a suitably chosen contour in the upper half plane or otherwise, evaluate the real integrals
and
where is taken to be the positive square root.
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Paper 3, Section II, F
2017 commentLet be an entire function. Prove Taylor's theorem, that there exist complex numbers such that for all . [You may assume Cauchy's Integral Formula.]
For a positive real , let . Explain why we have
for all .
Now let and be fixed. For which entire functions do we have
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Paper 1, Section I, A
2017 commentLet where . Suppose is an analytic function of in a domain of the complex plane.
Derive the Cauchy-Riemann equations satisfied by and .
For find a suitable function and domain such that is analytic in .
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Paper 1, Section II, A
2017 comment(a) Let be defined on the complex plane such that as and is analytic on an open set containing , where is a positive real constant.
Let be the horizontal contour running from to and let
By evaluating the integral, show that is analytic for .
(b) Let be defined on the complex plane such that as with . Suppose is analytic at all points except and which are simple poles with and .
Let be the horizontal contour running from to , and let
(i) Show that is analytic for .
(ii) Show that is analytic for .
(iii) Show that if then .
[You should be careful to make sure you consider all points in the required regions.]
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Paper 3, Section I, A
2017 commentBy using the Laplace transform, show that the solution to
subject to the conditions and , is given by
when .
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Paper 4, Section II, A
2017 commentBy using Fourier transforms and a conformal mapping
with and , and a suitable real constant , show that the solution to
is given by
where is to be determined.
In the case of , give explicitly as a function of . [You need not evaluate the integral.]
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Paper 2, Section I,
2017 commentState Gauss's Law in the context of electrostatics.
A spherically symmetric capacitor consists of two conductors in the form of concentric spherical shells of radii and , with . The inner sphere carries a charge and the outer sphere carries a charge . Determine the electric field and the electrostatic potential in the regions and . Show that the capacitance is
and calculate the electrostatic energy of the system in terms of and .
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Paper 4 , Section I,
2017 commentA thin wire, in the form of a closed curve , carries a constant current . Using either the Biot-Savart law or the magnetic vector potential, show that the magnetic field far from the loop is of the approximate form
where is the magnetic dipole moment of the loop. Derive an expression for in terms of and the vector area spanned by the curve .
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Paper 1, Section II, C
2017 commentWrite down Maxwell's equations for the electric field and the magnetic field in a vacuum. Deduce that both and satisfy a wave equation, and relate the wave speed to the physical constants and .
Verify that there exist plane-wave solutions of the form
where and are constant complex vectors, is a constant real vector and is a real constant. Derive the dispersion relation that relates the angular frequency of the wave to the wavevector , and give the algebraic relations between the vectors and implied by Maxwell's equations.
Let be a constant real unit vector. Suppose that a perfect conductor occupies the region with a plane boundary . In the vacuum region , a plane electromagnetic wave of the above form, with , is incident on the plane boundary. Write down the boundary conditions on and at the surface of the conductor. Show that Maxwell's equations and the boundary conditions are satisfied if the solution in the vacuum region is the sum of the incident wave given above and a reflected wave of the form
where
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Paper 3, Section II, C
2017 comment(i) Two point charges, of opposite sign and unequal magnitude, are placed at two different locations. Show that the combined electrostatic potential vanishes on a sphere that encloses only the charge of smaller magnitude.
(ii) A grounded, conducting sphere of radius is centred at the origin. A point charge is located outside the sphere at position vector . Formulate the differential equation and boundary conditions for the electrostatic potential outside the sphere. Using the result of part (i) or otherwise, show that the electric field outside the sphere is identical to that generated (in the absence of any conductors) by the point charge and an image charge located inside the sphere at position vector , provided that and are chosen correctly.
Calculate the magnitude and direction of the force experienced by the charge .
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Paper 2, Section II, C
2017 commentIn special relativity, the electromagnetic fields can be derived from a 4-vector potential . Using the Minkowski metric tensor and its inverse , state how the electromagnetic tensor is related to the 4-potential, and write out explicitly the components of both and in terms of those of and .
If is a Lorentz transformation of the spacetime coordinates from one inertial frame to another inertial frame , state how is related to .
Write down the Lorentz transformation matrix for a boost in standard configuration, such that frame moves relative to frame with speed in the direction. Deduce the transformation laws
where
In frame , an infinitely long wire of negligible thickness lies along the axis. The wire carries positive charges per unit length, which travel at speed in the direction, and negative charges per unit length, which travel at speed in the direction. There are no other sources of the electromagnetic field. Write down the electric and magnetic fields in in terms of Cartesian coordinates. Calculate the electric field in frame , which is related to by a boost by speed as described above. Give an explanation of the physical origin of your expression.
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Paper 1, Section I, D
2017 commentFor each of the flows
(i)
(ii)
determine whether or not the flow is incompressible and/or irrotational. Find the associated velocity potential and/or stream function when appropriate. For either one of the flows, sketch the streamlines of the flow, indicating the direction of the flow.
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Paper 2, Section I,
2017 commentFrom Euler's equations describing steady inviscid fluid flow under the action of a conservative force, derive Bernoulli's equation for the pressure along a streamline of the flow, defining all variables that you introduce.
Water fills an inverted, open, circular cone (radius increasing upwards) of half angle to a height above its apex. At time , the tip of the cone is removed to leave a small hole of radius . Assuming that the flow is approximately steady while the depth of water is much larger than , show that the time taken for the water to drain is approximately
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Paper 1, Section II, D
2017 commentA layer of thickness of fluid of density and dynamic viscosity flows steadily down and parallel to a rigid plane inclined at angle to the horizontal. Wind blows over the surface of the fluid and exerts a stress on the surface of the fluid in the upslope direction.
(a) Draw a diagram of this situation, including indications of the applied stresses and body forces, a suitable coordinate system and a representation of the expected velocity profile.
(b) Write down the equations and boundary conditions governing the flow, with a brief description of each, paying careful attention to signs. Solve these equations to determine the pressure and velocity fields.
(c) Determine the volume flux and show that there is no net flux if
Draw a sketch of the corresponding velocity profile.
(d) Determine the value of for which the shear stress on the rigid plane is zero and draw a sketch of the corresponding velocity profile.
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Paper 4, Section II, D
2017 commentThe linearised equations governing the horizontal components of flow in a rapidly rotating shallow layer of depth , where , are
where is the constant Coriolis parameter, and is the unit vector in the vertical direction.
Use these equations, either in vector form or using Cartesian components, to show that the potential vorticity
is independent of time, where is the relative vorticity.
Derive the equation
In the case that , determine and sketch the dispersion relation for plane waves with , where is constant. Discuss the nature of the waves qualitatively: do long waves propagate faster or slower than short waves; how does the phase speed depend on wavelength; does rotation have more effect on long waves or short waves; how dispersive are the waves?
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Paper 3, Section II, D
2017 commentUse Euler's equations to derive the vorticity equation
where is the fluid velocity and is the vorticity.
Consider axisymmetric, incompressible, inviscid flow between two rigid plates at and in cylindrical polar coordinates , where is time. Using mass conservation, or otherwise, find the complete flow field whose radial component is independent of .
Now suppose that the flow has angular velocity and that when . Use the vorticity equation to determine the angular velocity for subsequent times as a function of . What physical principle does your result illustrate?
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Paper 1, Section I, G
2017 commentGive the definition for the area of a hyperbolic triangle with interior angles .
Let . Show that the area of a convex hyperbolic -gon with interior angles is .
Show that for every and for every with there exists a regular hyperbolic -gon with area .
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Paper 3, Section I, G
2017 commentLet
be stereographic projection from the unit sphere in to the Riemann sphere . Show that if is a rotation of , then is a Möbius transformation of which can be represented by an element of . (You may assume without proof any result about generation of by a particular set of rotations, but should state it carefully.)
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Paper 2, Section II, G
2017 commentLet be a hyperplane in , where is a unit vector and is a constant. Show that the reflection map
is an isometry of which fixes pointwise.
Let be distinct points in . Show that there is a unique reflection mapping to , and that if and only if and are equidistant from the origin.
Show that every isometry of can be written as a product of at most reflections. Give an example of an isometry of which cannot be written as a product of fewer than 3 reflections.
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Paper 3, Section II, G
2017 commentLet be a parametrised surface, where is an open set.
(a) Explain what are the first and second fundamental forms of the surface, and what is its Gaussian curvature. Compute the Gaussian curvature of the hyperboloid .
(b) Let and be parametrised curves in , and assume that
Find a formula for the first fundamental form, and show that the Gaussian curvature vanishes if and only if
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Paper 4, Section II, G
2017 commentWhat is a hyperbolic line in (a) the disc model (b) the upper half-plane model of the hyperbolic plane? What is the hyperbolic distance between two points in the hyperbolic plane? Show that if is any continuously differentiable curve with endpoints and then its length is at least , with equality if and only if is a monotonic reparametrisation of the hyperbolic line segment joining and .
What does it mean to say that two hyperbolic lines are (a) parallel (b) ultraparallel? Show that and are ultraparallel if and only if they have a common perpendicular, and if so, then it is unique.
A horocycle is a curve in the hyperbolic plane which in the disc model is a Euclidean circle with exactly one point on the boundary of the disc. Describe the horocycles in the upper half-plane model. Show that for any pair of horocycles there exists a hyperbolic line which meets both orthogonally. For which pairs of horocycles is this line unique?
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Paper 3, Section I, E
2017 commentLet be a commutative ring and let be an -module. Show that is a finitely generated -module if and only if there exists a surjective -module homomorphism for some .
Find an example of a -module such that there is no surjective -module homomorphism but there is a surjective -module homomorphism which is not an isomorphism. Justify your answer.
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Paper 2, Section I, E
2017 comment(a) Define what is meant by a unique factorisation domain and by a principal ideal domain. State Gauss's lemma and Eisenstein's criterion, without proof.
(b) Find an example, with justification, of a ring and a subring such that
(i) is a principal ideal domain, and
(ii) is a unique factorisation domain but not a principal ideal domain.
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Paper 4, Section I,
2017 commentLet be a non-trivial finite -group and let be its centre. Show that . Show that if and if is not abelian, then .
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Paper 1, Section II, 10E
2017 comment(a) State Sylow's theorem.
(b) Let be a finite simple non-abelian group. Let be a prime number. Show that if divides , then divides where is the number of Sylow -subgroups of .
(c) Let be a group of order 48 . Show that is not simple. Find an example of which has no normal Sylow 2-subgroup.
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Paper 2, Section II, E
2017 commentLet be a commutative ring.
(a) Let be the set of nilpotent elements of , that is,
Show that is an ideal of .
(b) Assume is Noetherian and assume is a non-empty subset such that if , then . Let be an ideal of disjoint from . Show that there is a prime ideal of containing and disjoint from .
(c) Again assume is Noetherian and let be as in part (a). Let be the set of all prime ideals of . Show that
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Paper 4, Section II, E
2017 comment(a) State (without proof) the classification theorem for finitely generated modules over a Euclidean domain. Give the statement and the proof of the rational canonical form theorem.
(b) Let be a principal ideal domain and let be an -submodule of . Show that is a free -module.
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Paper 3, Section II, E
2017 comment(a) Define what is meant by a Euclidean domain. Show that every Euclidean domain is a principal ideal domain.
(b) Let be a prime number and let be a monic polynomial of positive degree. Show that the quotient ring is finite.
(c) Let and let be a non-zero prime ideal of . Show that the quotient is a finite ring.
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Paper 2, Section I, F
2017 commentState and prove the Rank-Nullity theorem.
Let be a linear map from to of rank 2 . Give an example to show that may be the direct sum of the kernel of and the image of , and also an example where this is not the case.
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Paper 1, Section I, F
2017 commentState and prove the Steinitz Exchange Lemma.
Deduce that, for a subset of , any two of the following imply the third:
(i) is linearly independent
(ii) is spanning
(iii) has exactly elements
Let be a basis of . For which values of do form a basis of
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Paper 4, Section I, F
2017 commentBriefly explain the Gram-Schmidt orthogonalisation process in a real finite-dimensional inner product space .
For a subspace of , define , and show that .
For which positive integers does
define an inner product on the space of all real polynomials of degree at most ?
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Paper 1, Section II, F
2017 commentLet and be finite-dimensional real vector spaces, and let be a surjective linear map. Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate.
(i) There is a linear map such that is the identity map on .
(ii) There is a linear map such that is the identity map on .
(iii) There is a subspace of such that the restriction of to is an isomorphism from to .
(iv) If and are subspaces of with then .
(v) If and are subspaces of with then .
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Paper 2, Section II, F
2017 commentLet and be linear maps between finite-dimensional real vector spaces.
Show that the rank satisfies . Show also that . For each of these two inequalities, give examples to show that we may or may not have equality.
Now let have dimension and let be a linear map of rank such that . Find the rank of for each .
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Paper 4, Section II, F
2017 commentWhat is the dual of a finite-dimensional real vector space ? If has a basis , define the dual basis, and prove that it is indeed a basis of .
[No results on the dimension of duals may be assumed without proof.]
Write down (without making a choice of basis) an isomorphism from to . Prove that your map is indeed an isomorphism.
Does every basis of arise as the dual basis of some basis of Justify your answer.
A subspace of is called separating if for every non-zero there is a with . Show that the only separating subspace of is itself.
Now let be the (infinite-dimensional) space of all real polynomials. Explain briefly how we may identify with the space of all real sequences. Give an example of a proper subspace of that is separating.
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Paper 3, Section II, F
2017 commentLet be a quadratic form on a finite-dimensional real vector space . Prove that there exists a diagonal basis for , meaning a basis with respect to which the matrix of is diagonal.
Define the rank and signature of in terms of this matrix. Prove that and are independent of the choice of diagonal basis.
In terms of , and the dimension of , what is the greatest dimension of a subspace on which is zero?
Now let be the quadratic form on given by . For which points in is it the case that there is some diagonal basis for containing ?
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Paper 3, Section I, H
2017 comment(a) What does it mean to say that a Markov chain is reversible?
(b) Let be a finite connected graph on vertices. What does it mean to say that is a simple random walk on ?
Find the unique invariant distribution of .
Show that is reversible when .
[You may use, without proof, results about detailed balance equations, but you should state them clearly.]
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Paper 4, Section I,
2017 commentProve that the simple symmetric random walk on is transient.
[Any combinatorial inequality can be used without proof.]
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Paper 1, Section II, H
2017 commentA rich and generous man possesses pounds. Some poor cousins arrive at his mansion. Being generous he decides to give them money. On day 1 , he chooses uniformly at random an integer between and 1 inclusive and gives it to the first cousin. Then he is left with pounds. On day 2 , he chooses uniformly at random an integer between and 1 inclusive and gives it to the second cousin and so on. If then he does not give the next cousin any money. His choices of the uniform numbers are independent. Let be his fortune at the end of day .
Show that is a Markov chain and find its transition probabilities.
Let be the first time he has 1 pound left, i.e. . Show that
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Paper 2, Section II, H
2017 commentLet be i.i.d. random variables with values in and . Moreover, suppose that the greatest common divisor of is 1 . Consider the following process
(a) Show that is a Markov chain and find its transition probabilities.
(b) Let . Find .
(c) Find the limit as of . State carefully any theorems from the course that you are using.
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Paper 2, Section I, B
2017 commentExpand as a Fourier series on .
By integrating the series show that on can be written as
where , should be determined and
By evaluating another way show that
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Paper 4, Section I, A
2017 commentThe Legendre polynomials, for integers , satisfy the Sturm-Liouville equation
and the recursion formula
(i) For all , show that is a polynomial of degree with .
(ii) For all , show that and are orthogonal over the range when .
(iii) For each let
Assume that for each there is a constant such that for all . Determine for each .
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Paper 3, Section I, A
2017 commentUsing the substitution , find that satisfies
with boundary data .
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Paper 1, Section II, 14B
2017 comment(a)
(i) Compute the Fourier transform of , where is a real positive constant.
(ii) Consider the boundary value problem
with real constant and boundary condition as .
Find the Fourier transform of and hence solve the boundary value problem. You should clearly state any properties of the Fourier transform that you use.
(b) Consider the wave equation
with initial conditions
Show that the Fourier transform of the solution with respect to the variable is given by
where and are the Fourier transforms of the initial conditions. Starting from derive d'Alembert's solution for the wave equation:
You should state clearly any properties of the Fourier transform that you use.
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Paper 3, Section II, A
2017 commentLet be the linear differential operator
where denotes differentiation with respect to .
Find the Green's function, , for satisfying the homogeneous boundary conditions .
Using the Green's function, solve
with boundary conditions . Here is the Heaviside step function having value 0 for and 1 for .
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Paper 2, Section II, A
2017 commentLaplace's equation for in cylindrical coordinates , is
Use separation of variables to find an expression for the general solution to Laplace's equation in cylindrical coordinates that is -periodic in .
Find the bounded solution that satisfies
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Paper 4, Section II, B
2017 comment(a)
(i) For the diffusion equation
with diffusion constant , state the properties (in terms of the Dirac delta function) that define the fundamental solution and the Green's function .
You are not required to give expressions for these functions.
(ii) Consider the initial value problem for the homogeneous equation:
and the forced equation with homogeneous initial condition (and given forcing term :
Given that and in part (i) are related by
(where is the Heaviside step function having value 0 for and 1 for , show how the solution of (B) can be expressed in terms of solutions of (A) with suitable initial conditions. Briefly interpret your expression.
(b) A semi-infinite conducting plate lies in the plane in the region . The boundary along the axis is perfectly insulated. Let denote standard polar coordinates on the plane. At time the entire plate is at temperature zero except for the region defined by and which has constant initial temperature . Subsequently the temperature of the plate obeys the two-dimensional heat equation with diffusion constant . Given that the fundamental solution of the twodimensional heat equation on is
show that the origin on the plate reaches its maximum temperature at time .
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Paper 3, Section I,
2017 commentLet and be topological spaces.
(a) Define what is meant by the product topology on . Define the projection maps and and show they are continuous.
(b) Consider in . Show that is Hausdorff if and only if is a closed subset of in the product topology.
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Paper 2, Section I, E
2017 commentLet be a function between metric spaces.
(a) Give the definition for to be continuous. Show that is continuous if and only if is an open subset of for each open subset of .
(b) Give an example of such that is not continuous but is an open subset of for every open subset of .
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Paper 1, Section II, E
2017 commentConsider and with their usual Euclidean topologies.
(a) Show that a non-empty subset of is connected if and only if it is an interval. Find a compact subset for which has infinitely many connected components.
(b) Let be a countable subset of . Show that is path-connected. Deduce that is not homeomorphic to .
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Paper 4, Section II, E
2017 commentLet be a continuous map between topological spaces.
(a) Assume is compact and that is a closed subset. Prove that and are both compact.
(b) Suppose that
(i) is compact for each , and
(ii) if is any closed subset of , then is a closed subset of .
Show that if is compact, then is compact.
Hint: Given an open cover , find a finite subcover, say , for each ; use closedness of and property (ii) to produce an open cover of .]
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Paper 1, Section I, C
2017 commentGiven real points , define the Lagrange cardinal polynomials . Let be the polynomial of degree that interpolates the function at these points. Express in terms of the values , and the Lagrange cardinal polynomials.
Define the divided difference and give an expression for it in terms of and . Prove that
for some number .
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Paper 4, Section I, C
2017 commentFor the matrix
find a factorization of the form
where is diagonal and is lower triangular with ones on its diagonal.
For what values of is positive definite?
In the case find the Cholesky factorization of .
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Paper 1, Section II, C
2017 commentA three-stage explicit Runge-Kutta method for solving the autonomous ordinary differential equation
is given by
where
and is the time-step. Derive sufficient conditions on the coefficients , and for the method to be of third order.
Assuming that these conditions hold, verify that belongs to the linear stability domain of the method.
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Paper 2, Section II, C
2017 commentDefine the linear least-squares problem for the equation , where is an matrix with is a given vector and is an unknown vector.
If , where is an orthogonal matrix and is an upper triangular matrix in standard form, explain why the least-squares problem is solved by minimizing the Euclidean norm .
Using the method of Householder reflections, find a QR factorization of the matrix
Hence find the solution of the least-squares problem in the case
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Paper 3, Section II, C
2017 commentLet be the th monic orthogonal polynomial with respect to the inner product
on , where is a positive weight function.
Prove that, for has distinct zeros in the interval .
Let be distinct points. Show that the quadrature formula
is exact for all if the weights are chosen to be
Show further that the quadrature formula is exact for all if the nodes are chosen to be the zeros of (Gaussian quadrature). [Hint: Write as , where .]
Use the Peano kernel theorem to write an integral expression for the approximation error of Gaussian quadrature for sufficiently differentiable functions. (You should give a formal expression for the Peano kernel but are not required to evaluate it.)
-
Paper 1, Section I, H
2017 commentSolve the following linear programming problem using the simplex method:
Suppose we now subtract from the right hand side of the last two constraints. Find the new optimal value.
-
Paper 2, Section I, H
2017 commentConsider the following optimization problem
(a) Write down the Lagrangian for this problem. State the Lagrange sufficiency theorem.
(b) Formulate the dual problem. State and prove the weak duality property.
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Paper 4, Section II, H
2017 comment(a) Let be a flow network with capacities on the edges. Explain the maximum flow problem on this network defining all the notation you need.
(b) Describe the Ford-Fulkerson algorithm for finding a maximum flow and state the max-flow min-cut theorem.
(c) Apply the Ford-Fulkerson algorithm to find a maximum flow and a minimum cut of the following network:

(d) Suppose that we add to each capacity of a flow network. Is it true that the maximum flow will always increase by ? Justify your answer.
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Paper 3, Section II, H
2017 comment(a) Explain what is meant by a two-person zero-sum game with payoff matrix and define what is an optimal strategy (also known as a maximin strategy) for each player.
(b) Suppose the payoff matrix is antisymmetric, i.e. and for all . What is the value of the game? Justify your answer.
(c) Consider the following two-person zero-sum game. Let . Both players simultaneously call out one of the numbers . If the numbers differ by one, the player with the higher number wins from the other player. If the players' choices differ by 2 or more, the player with the higher number pays to the other player. In the event of a tie, no money changes hands.
Write down the payoff matrix.
For the case when find the value of the game and an optimal strategy for each player.
Find the value of the game and an optimal strategy for each player for all .
[You may use results from the course provided you state them clearly.]
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Paper 4, Section I, B
2017 comment(a) Give a physical interpretation of the wavefunction (where and are real constants).
(b) A particle of mass and energy is incident from the left on the potential step
with .
State the conditions satisfied by a stationary state at the point .
Compute the probability that the particle is reflected as a function of , and compare your result with the classical case.
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Paper 3, Section I, B
2017 commentA particle of mass is confined to a one-dimensional box . The potential is zero inside the box and infinite outside.
(a) Find the allowed energies of the particle and the normalised energy eigenstates.
(b) At time the particle has wavefunction that is uniform in the left half of the box i.e. for and for . Find the probability that a measurement of energy at time will yield a value less than .
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Paper 1, Section II, B
2017 commentConsider the time-independent Schrödinger equation in one dimension for a particle of mass with potential .
(a) Show that if the potential is an even function then any non-degenerate stationary state has definite parity.
(b) A particle of mass is subject to the potential given by
where and are real positive constants and is the Dirac delta function.
Derive the conditions satisfied by the wavefunction around the points .
Show (using a graphical method or otherwise) that there is a bound state of even parity for any , and that there is an odd parity bound state only if . [Hint: You may assume without proof that the functions and are monotonically increasing for .]
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Paper 3, Section II, B
2017 comment(a) Given the position and momentum operators and (for in three dimensions, define the angular momentum operators and the total angular momentum .
Show that is Hermitian.
(b) Derive the generalised uncertainty relation for the observables and in the form
for any state and a suitable expression that you should determine. [Hint: It may be useful to consider the operator .]
(c) Consider a particle with wavefunction
where and and are real positive constants.
Show that is an eigenstate of total angular momentum and find the corresponding angular momentum quantum number . Find also the expectation value of a measurement of on the state .
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Paper 2, Section II, B
2017 comment(a) The potential for the one-dimensional harmonic oscillator is . By considering the associated time-independent Schrödinger equation for the wavefunction with substitutions
show that the allowed energy levels are given by for [You may assume without proof that must be a polynomial for to be normalisable.]
(b) Consider a particle with charge and mass subject to the one-dimensional harmonic oscillator potential . You may assume that the normalised ground state of this potential is
The particle is in the stationary state corresponding to when at time , an electric field of constant strength is turned on, adding an extra term to the harmonic potential.
(i) Using the result of part (a) or otherwise, find the energy levels of the new potential.
(ii) Show that the probability of finding the particle in the ground state immediately after is given by . [You may assume that .]
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Paper 1, Section I, H
2017 comment(a) State and prove the Rao-Blackwell theorem.
(b) Let be an independent sample from with to be estimated. Show that is an unbiased estimator of and that is a sufficient statistic.
What is
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Paper 2, Section I, 8H
2017 comment(a) Define a confidence interval for an unknown parameter .
(b) Let be i.i.d. random variables with distribution with unknown. Find a confidence interval for .
[You may use the fact that
(c) Let be independent with to be estimated. Find a confidence interval for .
Suppose that we have two observations and . What might be a better interval to report in this case?
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Paper 4, Section II, H
2017 comment(a) State and prove the Neyman-Pearson lemma.
(b) Let be a real random variable with density with
Find a most powerful test of size of versus .
Find a uniformly most powerful test of size of versus .
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Paper 1, Section II, H
2017 comment(a) Give the definitions of a sufficient and a minimal sufficient statistic for an unknown parameter .
Let be an independent sample from the geometric distribution with success probability and mean , i.e. with probability mass function
Find a minimal sufficient statistic for . Is your statistic a biased estimator of
[You may use results from the course provided you state them clearly.]
(b) Define the bias of an estimator. What does it mean for an estimator to be unbiased?
Suppose that has the truncated Poisson distribution with probability mass function
Show that the only unbiased estimator of based on is obtained by taking if is odd and if is even.
Is this a useful estimator? Justify your answer.
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Paper 3, Section II,
2017 commentConsider the general linear model
where is a known matrix of full rank with known and is an unknown vector.
(a) State without proof the Gauss-Markov theorem.
Find the maximum likelihood estimator for . Is it unbiased?
Let be any unbiased estimator for which is linear in . Show that
for all .
(b) Suppose now that and that and are both unknown. Find the maximum likelihood estimator for . What is the joint distribution of and in this case? Justify your answer.
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Paper 1, Section I, D
2017 commentDerive the Euler-Lagrange equation for the function that gives a stationary value of
where is a bounded domain in the -plane and is fixed on the boundary .
Find the equation satisfied by the function that gives a stationary value of
where is a constant and is prescribed on .
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Paper 3, Section , D
2017 comment(a) A Pringle crisp can be defined as the surface
Use the method of Lagrange multipliers to find the minimum and maximum values of on the boundary of the Pringle crisp and the positions where these occur.
(b) A farmer wishes to construct a grain silo in the form of a hollow vertical cylinder of radius and height with a hollow hemispherical cap of radius on top of the cylinder. The walls of the cylinder cost per unit area to construct and the surface of the cap costs per unit area to construct. Given that a total volume is desired for the silo, what values of and should be chosen to minimise the cost?
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Paper 2, Section II, D
2017 commentA proto-planet of mass in a uniform galactic dust cloud has kinetic and potential energies
where is constant. State Hamilton's principle and use it to determine the equations of motion for the proto-planet.
Write down two conserved quantities of the motion and state why their existence illustrates Noether's theorem.
Determine the Hamiltonian of this system, where and are the conjugate momenta corresponding to .
Write down Hamilton's equations for this system and use them to show that
and is a constant. With the aid of a diagram, explain why there is a stable circular orbit.
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Paper 4, Section II,
2017 commentConsider the functional
of a function defined for , with having fixed values at and .
By considering , where is an arbitrary function with and , determine that the second variation of is
The surface area of an axisymmetric soap film joining two parallel, co-axial, circular rings of radius a distance apart can be expressed by the functional
where is distance in the axial direction and is radial distance from the axis. Show that the surface area is stationary when
where is a constant that need not be determined, and that the stationary area is a local minimum if
for all functions that vanish at , where .
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Paper 2, Section II, I
2017 commentLet be an algebraically closed field of any characteristic.
(a) Define what it means for a variety to be non-singular at a point .
(b) Let be a hypersurface for an irreducible homogeneous polynomial. Show that the set of singular points of is , where is the ideal generated by
(c) Consider the projective plane curve corresponding to the affine curve in given by the equation
Find the singular points of this projective curve if char . What goes wrong if char ?
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Paper 3, Section II, I
2017 comment(a) Define what it means to give a rational map between algebraic varieties. Define a birational map.
(b) Let
Define a birational map from to . [Hint: Consider lines through the origin.]
(c) Let be the surface given by the equation
Consider the blow-up of at the origin, i.e. the subvariety of defined by the equations for , with coordinates on . Let be the projection and . Recall that the proper transform of is the closure of in . Give equations for , and describe the fibres of the morphism .
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Paper 4, Section II, I
2017 comment(a) Let and be non-singular projective curves over a field and let be a non-constant morphism. Define the ramification degree of at a point .
(b) Suppose char . Let be the plane cubic with , and let . Explain how the projection
defines a morphism . Determine the degree of and the ramification degrees for all .
(c) Let be a non-singular projective curve and let . Show that there is a non-constant rational function on which is regular on .
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Paper 1, Section II, I
2017 commentLet be an algebraically closed field.
(a) Let and be varieties defined over . Given a function , define what it means for to be a morphism of varieties.
(b) If is an affine variety, show that the coordinate ring coincides with the ring of regular functions on . [Hint: You may assume a form of the Hilbert Nullstellensatz.]
(c) Now suppose and are affine varieties. Show that if and are isomorphic, then there is an isomorphism of -algebras .
(d) Show that is not isomorphic to .
-
Paper 3, Section II, I
2017 commentThe -torus is the product of circles:
For all and , compute .
[You may assume that relevant spaces are triangulable, but you should state carefully any version of any theorem that you use.]
-
Paper 2, Section II, I
2017 comment(a) (i) Define the push-out of the following diagram of groups.

When is a push-out a free product with amalgamation?
(ii) State the Seifert-van Kampen theorem.
(b) Let (recalling that is the real projective plane), and let .
(i) Compute the fundamental group of the space .
(ii) Show that there is a surjective homomorphism , where is the symmetric group on three elements.
(c) Let be the covering space corresponding to the kernel of .
(i) Draw and justify your answer carefully.
(ii) Does retract to a graph? Justify your answer briefly.
(iii) Does deformation retract to a graph? Justify your answer briefly.
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Paper 1, Section II, I
2017 commentLet be a topological space and let and be points of .
(a) Explain how a path from to defines a map .
(b) Prove that is an isomorphism of groups.
(c) Let be based loops in . Suppose that are homotopic as unbased maps, i.e. the homotopy is not assumed to respect basepoints. Show that the corresponding elements of are conjugate.
(d) Take to be the 2-torus . If are homotopic as unbased loops as in part (c), then exhibit a based homotopy between them. Interpret this fact algebraically.
(e) Exhibit a pair of elements in the fundamental group of which are homotopic as unbased loops but not as based loops. Justify your answer.
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Paper 4, Section II, I
2017 commentRecall that is real projective -space, the quotient of obtained by identifying antipodal points. Consider the standard embedding of as the unit sphere in .
(a) For odd, show that there exists a continuous map such that is orthogonal to , for all .
(b) Exhibit a triangulation of .
(c) Describe the map induced by the antipodal map, justifying your answer.
(d) Show that, for even, there is no continuous map such that is orthogonal to for all .
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Paper 3, Section II, F
2017 commentDenote by the space of continuous complex-valued functions on converging to zero at infinity. Denote by the Fourier transform of .
(i) Prove that the image of under is included and dense in , and that is injective. [Fourier inversion can be used without proof when properly stated.]
(ii) Calculate the Fourier transform of , the characteristic function of .
(iii) Prove that belongs to and is the Fourier transform of a function , which you should determine.
(iv) Using the functions and the open mapping theorem, deduce that the Fourier transform is not surjective from to .
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Paper 4, Section II,
2017 commentConsider with the Lebesgue measure. Denote by the Fourier transform of and by the Fourier-Plancherel transform of . Let for and define for
(i) Prove that is a vector subspace of , and is a Hilbert space for the inner product , where denotes the complex conjugate of .
(ii) Construct a function , that is not almost everywhere equal to a continuous function.
(iii) For , prove that is a well-defined function and that converges to in as .
[Hint: Prove that where is an approximation of the unit as
(iv) Deduce that if and then .
[Hint: Prove that: (1) there is a sequence such that converges to almost everywhere; (2) is uniformly bounded in as .]
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Paper 1, Section II,
2017 commentConsider a sequence of measurable functions converging pointwise to a function . The Lebesgue measure is denoted by .
(a) Consider a Borel set with finite Lebesgue measure . Define for the sets
Prove that for any , one has and . Prove that for any .
(b) Consider a Borel set with finite Lebesgue measure . Prove that for any , there is a Borel set for which and such that converges to uniformly on as . Is the latter still true when ?
(c) Assume additionally that for some , and there exists an for which for all . Prove that .
(d) Let and be as in part (c). Consider a Borel set with finite Lebesgue measure . Prove that are integrable on and as . Deduce that converges weakly to in when . Does the convergence have to be strong?
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Paper 1, Section II, C
2017 commentA one-dimensional lattice has sites with lattice spacing . In the tight-binding approximation, the Hamiltonian describing a single electron is given by
where is the normalised state of the electron localised on the lattice site. Using periodic boundary conditions , solve for the spectrum of this Hamiltonian to derive the dispersion relation
Define the Brillouin zone. Determine the number of states in the Brillouin zone.
Calculate the velocity and effective mass of the particle. For which values of is the effective mass negative?
In the semi-classical approximation, derive an expression for the time-dependence of the position of the electron in a constant electric field.
Describe how the concepts of metals and insulators arise in the model above.
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Paper 2, Section II, C
2017 commentGive an account of the variational method for establishing an upper bound on the ground-state energy of a Hamiltonian with a discrete spectrum , where
A particle of mass moves in the three-dimensional potential
where are constants and is the distance to the origin. Using the normalised variational wavefunction
show that the expected energy is given by
Explain why there is necessarily a bound state when . What can you say about the existence of a bound state when ?
[Hint: The Laplacian in spherical polar coordinates is
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Paper 3, Section II, C
2017 commentA particle of mass and charge moving in a uniform magnetic field is described by the Hamiltonian
where is the canonical momentum, which obeys . The mechanical momentum is defined as . Show that
Define
Derive the commutation relation obeyed by and . Write the Hamiltonian in terms of and and hence solve for the spectrum.
In symmetric gauge, states in the lowest Landau level with have wavefunctions
where and is a positive integer. By considering the profiles of these wavefunctions, estimate how many lowest Landau level states can fit in a disc of radius .
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Paper 4, Section II, C
2017 comment(a) In one dimension, a particle of mass is scattered by a potential where as . For wavenumber , the incoming and outgoing asymptotic plane wave states with positive and negative parity are given by
(i) Explain how this basis may be used to define the -matrix,
(ii) For what choice of potential would you expect ? Why?
(b) The potential is given by
with a constant.
(i) Show that
where . Verify that . Explain the physical meaning of this result.
(ii) For , by considering the poles or zeros of , show that there exists one bound state of negative parity if .
(iii) For and , show that has a pole at
where and are real and
Explain the physical significance of this result.
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Paper 2, Section II, K
2017 comment(a) Give the definition of a Poisson process on . Let be a Poisson process on . Show that conditional on , the jump times have joint density function
where is the indicator of the set .
(b) Let be a Poisson process on with intensity and jump times . If is a real function, we define for all
Show that for all the following is true
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Paper 3, Section II, K
2017 comment(a) Define the Moran model and Kingman's -coalescent. Define Kingman's infinite coalescent.
Show that Kingman's infinite coalescent comes down from infinity. In other words, with probability one, the number of blocks of is finite at any time .
(b) Give the definition of a renewal process.
Let denote the sequence of inter-arrival times of the renewal process . Suppose that .
Prove that as .
Prove that for some strictly positive .
[Hint: Consider the renewal process with inter-arrival times for some suitable .]
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Paper 4, Section II,
2017 comment(a) Give the definition of an queue. Prove that if is the arrival rate and the service rate and , then the length of the queue is a positive recurrent Markov chain. What is the equilibrium distribution?
If the queue is in equilibrium and a customer arrives at some time , what is the distribution of the waiting time (time spent waiting in the queue plus service time)?
(b) We now modify the above queue: on completion of service a customer leaves with probability , or goes to the back of the queue with probability . Find the distribution of the total time a customer spends being served.
Hence show that equilibrium is possible if and find the stationary distribution.
Show that, in equilibrium, the departure process is Poisson.
[You may use relevant theorems provided you state them clearly.]
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Paper 1, Section II,
2017 comment(a) Define a continuous time Markov chain with infinitesimal generator and jump chain .
(b) Let be a transient state of a continuous-time Markov chain with . Show that the time spent in state has an exponential distribution and explicitly state its parameter.
[You may use the fact that if , then for .]
(c) Let be an asymmetric random walk in continuous time on the non-negative integers with reflection at 0 , so that
Suppose that and . Show that for all , the total time spent in state is exponentially distributed with parameter .
Assume now that has some general distribution with probability generating function . Find the expected amount of time spent at 0 in terms of .
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Paper 2, Section II, E
2017 commentConsider the function
where the contour is the boundary of the half-strip and , taken anti-clockwise.
Use integration by parts and the method of stationary phase to:
(i) Obtain the leading term for coming from the vertical lines for large .
(ii) Show that the leading term in the asymptotic expansion of the function for large positive is
and obtain an estimate for the remainder as for some to be determined.
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Paper 3, Section II, E
2017 commentConsider the integral representation for the modified Bessel function
where is a simple closed contour containing the origin, taken anti-clockwise.
Use the method of steepest descent to determine the full asymptotic expansion of for large real positive
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Paper 4, Section II, E
2017 commentConsider solutions to the equation
of the form
with the assumption that, for large positive , the function is small compared to for all
Obtain equations for the , which are formally equivalent to ( . Solve explicitly for and . Show that it is consistent to assume that for some constants . Give a recursion relation for the .
Deduce that there exist two linearly independent solutions to with asymptotic expansions as of the form
Determine a recursion relation for the . Compute and .
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Paper 1, Section I,
2017 comment(a) Prove that every regular language is also a context-free language (CFL).
(b) Show that, for any fixed , the set of regular expressions over the alphabet is a CFL, but not a regular language.
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Paper 2, Section I,
2017 comment(a) Give explicit examples, with justification, of a language over some finite alphabet which is:
(i) context-free, but not regular;
(ii) recursive, but not context-free.
(b) Give explicit examples, with justification, of a subset of which is:
(i) recursively enumerable, but not recursive;
(ii) neither recursively enumerable, nor having recursively enumerable complement in .
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Paper 3, Section I, 4H
2017 comment(a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form (CNF). Give an example, with justification, of a context-free language (CFL) which is not defined by any CFG in CNF.
(b) Show that the intersection of two CFLs need not be a CFL.
(c) Let be a CFL over an alphabet . Show that need not be a CFL.
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Paper 4, Section I,
2017 comment(a) Describe the process for converting a deterministic finite-state automaton into a regular expression defining the same language, . [You need only outline the steps, without proof, but you should clearly define all terminology you introduce.]
(b) Consider the language over the alphabet defined via
Show that satisfies the pumping lemma for regular languages but is not a regular language itself.
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Paper 1, Section II,
2017 comment(a) Give an encoding to integers of all deterministic finite-state automata (DFAs). [Here the alphabet of each such DFA is always taken from the set , and the states for each such DFA are always taken from the set
(b) Show that the set of codes for which the corresponding DFA accepts a finite language is recursive. Moreover, if the language is finite, show that we can compute its size from .
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Paper 3, Section II, Automata and formal languages
2017 comment(a) Given , define a many-one reduction of to . Show that if is recursively enumerable (r.e.) and then is also recursively enumerable.
(b) State the theorem, and use it to prove that a set is r.e. if and only if .
(c) Consider the sets of integers defined via
Show that .
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Paper 1, Section I, E
2017 commentConsider a Lagrangian system with Lagrangian , where , and constraints
Use the method of Lagrange multipliers to show that this is equivalent to a system with Lagrangian , where , and are coordinates on the surface of constraints.
Consider a bead of unit mass in constrained to move (with no potential) on a wire given by an equation , where are Cartesian coordinates. Show that the Euler-Lagrange equations take the form
for some which should be specified. Find one first integral of the EulerLagrange equations, and thus show that
where should be given in the form of an integral.
[Hint: You may assume that the Euler-Lagrange equations hold in all coordinate systems.]
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Paper 2, Section I, E
2017 commentDerive the Lagrange equations from the principle of stationary action
where the end points and are fixed.
Let and be a scalar and a vector, respectively, depending on . Consider the Lagrangian
and show that the resulting Euler-Lagrange equations are invariant under the transformations
where is an arbitrary function, and is a constant which should be determined.
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Paper 3, Section I, E
2017 commentDefine an integrable system with -dimensional phase space. Define angle-action variables.
Consider a two-dimensional phase space with the Hamiltonian
where is a positive integer and the mass changes slowly in time. Use the fact that the action is an adiabatic invariant to show that the energy varies in time as , where is a constant which should be found.
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Paper 4, Section I, E
2017 commentConsider the Poisson bracket structure on given by
and show that , where and is any polynomial function on .
Let , where are positive constants. Find the explicit form of Hamilton's equations
Find a condition on such that the oscillation described by
is linearly unstable, where are small.
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Paper 2, Section II, E
2017 commentShow that an object's inertia tensor about a point displaced from the centre of mass by a vector is given by
where is the total mass of the object, and is the inertia tensor about the centre of mass.
Find the inertia tensor of a cube of uniform density, with edge of length , about one of its vertices.
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Paper 4, Section II,
2017 commentExplain how geodesics of a Riemannian metric
arise from the kinetic Lagrangian
where .
Find geodesics of the metric on the upper half plane
with the metric
and sketch the geodesic containing the points and .
[Hint: Consider
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Paper 1, Section I, G
2017 commentLet be a binary code of length . Define the following decoding rules: (i) ideal observer, (ii) maximum likelihood, (iii) minimum distance.
Let denote the probability that a digit is mistransmitted and suppose . Prove that maximum likelihood and minimum distance decoding agree.
Suppose codewords 000 and 111 are sent with probabilities and respectively with error probability . If we receive 110 , how should it be decoded according to the three decoding rules above?
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Paper 2, Section , G
2017 commentProve that a decipherable code with prescribed word lengths exists if and only if there is a prefix-free code with the same word lengths.
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Paper 3, Section I, G
2017 commentFind and describe all binary cyclic codes of length 7 . Pair each code with its dual code. Justify your answer.
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Paper 4, Section I, G
2017 commentDescribe the RSA system with public key and private key .
Give a simple example of how the system is vulnerable to a homomorphism attack.
Describe the El-Gamal signature scheme and explain how this can defeat a homomorphism attack.
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Paper 1, Section II, G
2017 commentLet be a binary linear code. Explain what it means for to have length and . Explain what it means for a codeword of to have weight .
Suppose has length , rank , and codewords of weight . The weight enumerator polynomial of is given by
What is Prove that if and only if .
Define the dual code of .
(i) Let . Show that
(ii) Extend the definition of weight to give a weight for . Suppose that for real and all
For real, by evaluating
in two different ways, show that
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Paper 2, Section II, G
2017 commentDefine the entropy, , of a random variable . State and prove Gibbs' inequality.
Hence, or otherwise, show that and determine when equality occurs.
Show that the Discrete Memoryless Channel with channel matrix
has capacity .
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Paper 1, Section I, C
2017 commentIn a homogeneous and isotropic universe, describe the relative displacement of two galaxies in terms of a scale factor . Show how the relative velocity of these galaxies is given by the relation , where you should specify in terms of .
From special relativity, the Doppler shift of light emitted by a particle moving away radially with speed can be approximated by
where is the wavelength of emitted light and is the observed wavelength. For the observed light from distant galaxies in a homogeneous and isotropic expanding universe, show that the redshift defined by is given by
where is the time of emission and is the observation time.
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Paper 2, Section I, C
2017 commentIn a homogeneous and isotropic universe , the acceleration equation for the scale factor is given by
where is the mass density and is the pressure.
If the matter content of the universe obeys the strong energy condition , show that the acceleration equation can be rewritten as , with Hubble parameter . Show that
where is the measured value today at . Hence, or otherwise, show that
Use this inequality to find an upper bound on the age of the universe.
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Paper 3, Section I, C
2017 comment(a) In the early universe electrons, protons and neutral hydrogen are in thermal equilibrium and interact via,
The non-relativistic number density of particles in thermal equlibrium is
where, for each species is the number of degrees of freedom, is its mass, and is its chemical potential. [You may assume and .]
Stating any assumptions required, use these expressions to derive the Saha equation which governs the relative abundances of electrons, protons and hydrogen,
where is the binding energy of hydrogen, which should be defined.
(b) Naively, we might expect that the majority of electrons and protons combine to form neutral hydrogen once the temperature drops below the binding energy, i.e. . In fact recombination does not happen until a much lower temperature, when . Briefly explain why this is.
[Hint: It may help to consider the relative abundances of particles in the early universe.]
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Paper 4, Section I, C
2017 comment(a) By considering a spherically symmetric star in hydrostatic equilibrium derive the pressure support equation
where is the radial distance from the centre of the star, is the stellar mass contained inside that radius, and and are the pressure and density at radius respectively.
(b) Propose, and briefly justify, boundary conditions for this differential equation, both at the centre of the star , and at the stellar surface .
Suppose that for some . Show that the density satisfies the linear differential equation
where , for some constant , is a rescaled radial coordinate. Find .
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Paper 3, Section II, C
2017 comment(a) The scalar moment of inertia for a system of particles is given by
where is the particle's mass and is a vector giving the particle's position. Show that, for non-relativistic particles,
where is the total kinetic energy of the system and is the total force on particle
Assume that any two particles and interact gravitationally with potential energy
Show that
where is the total potential energy of the system. Use the above to prove the virial theorem.
(b) Consider an approximately spherical overdensity of stationary non-interacting massive particles with initial constant density and initial radius . Assuming the system evolves until it reaches a stable virial equilibrium, what will the final and be in terms of their initial values? Would this virial solution be stable if our particles were baryonic rather than non-interacting? Explain your answer.
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Paper 1, Section II, C
2017 commentThe evolution of a flat homogeneous and isotropic universe with scale factor , mass density and pressure obeys the Friedmann and energy conservation equations
where is the Hubble parameter (observed today with value ) and is the cosmological constant.
Use these two equations to derive the acceleration equation
For pressure-free matter and , solve the energy conservation equation to show that the Friedmann and acceleration equations can be re-expressed as
where we have taken and we have defined the relative densities today as
Solve the Friedmann equation and show that the scale factor can be expressed as
Find an expression for the time at which the matter density and the effective density caused by the cosmological constant are equal. (You need not evaluate this explicitly.)
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Paper 2, Section II, I
2017 commentLet be a regular smooth curve. Define the curvature and torsion of and derive the Frenet formulae. Give the assumption which must hold for torsion to be well-defined, and state the Fundamental Theorem for curves in .
Let be as above and be another regular smooth curve with curvature and torsion . Suppose and for all , and that there exists a non-empty open subinterval such that . Show that .
Now let be an oriented surface and let be a regular smooth curve contained in . Define normal curvature and geodesic curvature. When is a geodesic? Give an example of a surface and a geodesic whose normal curvature vanishes identically. Must such a surface contain a piece of a plane? Can such a geodesic be a simple closed curve? Justify your answers.
Show that if is a geodesic and the Gaussian curvature of satisfies , then we have the inequality , where denotes the mean curvature of and the curvature of . Give an example where this inequality is sharp.
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Paper 3, Section II, I
2017 commentLet be a manifold and let be a smooth regular curve on . Define the total length and the arc length parameter . Show that can be reparametrized by arc length.
Let denote a regular surface, let be distinct points and let be a smooth regular curve such that . We say that is length minimising if for all smooth regular curves with , we have . By deriving a formula for the derivative of the energy functional corresponding to a variation of , show that a length minimising curve is necessarily a geodesic. [You may use the following fact: given a smooth vector field along with , there exists a variation of such that
Let denote the unit sphere and let denote the surface . For which pairs of points does there exist a length minimising smooth regular curve with and ? Justify your answer.
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Paper 4, Section II, I
2017 commentLet be a surface and . Define the exponential map exp and compute its differential . Deduce that is a local diffeomorphism.
Give an example of a surface and a point for which the exponential map fails to be defined globally on . Can this failure be remedied by extending the surface? In other words, for any such , is there always a surface such that the exponential map defined with respect to is globally defined on ?
State the version of the Gauss-Bonnet theorem with boundary term for a surface and a closed disc whose boundary can be parametrized as a smooth closed curve in .
Let be a flat surface, i.e. . Can there exist a closed disc , whose boundary can be parametrized as a smooth closed curve, and a surface such that all of the following hold:
(i) ;
(ii) letting be , we have that is a closed disc in with boundary
(iii) the Gaussian curvature of satisfies , and there exists a such that ?
Justify your answer.
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Paper 1, Section II, I
2017 commentDefine what it means for a subset to be a manifold.
For manifolds and , state what it means for a map to be smooth. For such a smooth map, and , define the differential map .
What does it mean for to be a regular value of ? Give an example of a map and a which is not a regular value of .
Show that the set of real-valued matrices with determinant 1 can naturally be viewed as a manifold . What is its dimension? Show that matrix multiplication , defined by , is smooth. [Standard theorems may be used without proof if carefully stated.] Describe the tangent space of at the identity as a subspace of .
Show that if then the set of real-valued matrices with determinant 0 , viewed as a subset of , is not a manifold.
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Paper 1, Section II, A
2017 commentConsider the dynamical system
where is a constant.
(a) Find the fixed points of the system, and classify them for .
Sketch the phase plane for each of the cases (i) (ii) and (iii) .
(b) Given , show that the domain of stability of the origin includes the union over of the regions
By considering , or otherwise, show that more information is obtained from the union over than considering only the case .
Hint: If then
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Paper 2, Section II, A
2017 comment(a) State Liapunov's first theorem and La Salle's invariance principle. Use these results to show that the fixed point at the origin of the system
is asymptotically stable.
(b) State the Poincaré-Bendixson theorem. Show that the forced damped pendulum
with , has a periodic orbit that encircles the cylindrical phase space , and that it is unique.
[You may assume that the Poincaré-Bendixson theorem also holds on a cylinder, and comment, without proof, on the use of any other standard results.]
(c) Now consider for , where . Use the energy-balance method to show that there is a homoclinic orbit in if , where .
Explain briefly why there is no homoclinic orbit in for .
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Paper 3, Section II, A
2017 commentState, without proof, the centre manifold theorem. Show that the fixed point at the origin of the system
where is a constant, is nonhyperbolic at . What are the dimensions of the linear stable and (non-extended) centre subspaces at this point?
Make the substitutions and and derive the resultant equations for and .
The extended centre manifold is given by
where and can be expanded as power series about . What is known about and from the centre manifold theorem? Assuming that , determine to and to . Hence obtain the evolution equation on the centre manifold correct to , and identify the type of bifurcation distinguishing between the cases and .
If now , assume that and extend your calculations of to and of the dynamics on the centre manifold to . Hence sketch the bifurcation diagram in the neighbourhood of .
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Paper 4, Section II, A
2017 commentConsider the one-dimensional map defined by
where and are constants, is a parameter and .
(a) Find the fixed points of and determine the linear stability of . Hence show that there are bifurcations at , at and, if , at .
Sketch the bifurcation diagram for each of the cases:
In each case show the locus and stability of the fixed points in the -plane, and state the type of each bifurcation. [Assume that there are no further bifurcations in the region sketched.]
(b) For the case (i.e. , you may assume that
Show that there are at most three 2-cycles and determine when they exist. By considering , or otherwise, show further that one 2-cycle is always unstable when it exists and that the others are unstable when . Sketch the bifurcation diagram showing the locus and stability of the fixed points and 2-cycles. State briefly what you would expect to occur for .
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Paper 1, Section II, 35D
2017 commentIn some inertial reference frame , there is a uniform electric field directed along the positive -direction and a uniform magnetic field directed along the positive direction. The magnitudes of the fields are and , respectively, with . Show that it is possible to find a reference frame in which the electric field vanishes, and determine the relative speed of the two frames and the magnitude of the magnetic field in the new frame.
[Hint: You may assume that under a standard Lorentz boost with speed c along the -direction, the electric and magnetic field components transform as
where the Lorentz factor .]
A point particle of mass and charge moves relativistically under the influence of the fields and . The motion is in the plane . By considering the motion in the reference frame in which the electric field vanishes, or otherwise, show that, with a suitable choice of origin, the worldline of the particle has components in the frame of the form
Here, is a constant speed with Lorentz factor is the particle's proper time, and is a frequency that you should determine.
Using dimensionless coordinates,
sketch the trajectory of the particle in the -plane in the limiting cases and .
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Paper 3, Section II, D
2017 commentBy considering the force per unit volume on a charge density and current density due to an electric field and magnetic field , show that
where and the symmetric tensor should be specified.
Give the physical interpretation of and and explain how can be used to calculate the net electromagnetic force exerted on the charges and currents within some region of space in static situations.
The plane carries a uniform charge per unit area and a current per unit length along the -direction. The plane carries the opposite charge and current. Show that between these planes
and for and .
Use to find the electromagnetic force per unit area exerted on the charges and currents in the plane. Show that your result agrees with direct calculation of the force per unit area based on the Lorentz force law.
If the current is due to the motion of the charge with speed , is it possible for the force between the planes to be repulsive?
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Paper 4, Section II, D
2017 commentA dielectric material has a real, frequency-independent relative permittivity with . In this case, the macroscopic polarization that develops when the dielectric is placed in an external electric field is . Explain briefly why the associated bound current density is
[You should ignore any magnetic response of the dielectric.]
A sphere of such a dielectric, with radius , is centred on . The sphere scatters an incident plane electromagnetic wave with electric field
where and is a constant vector. Working in the Lorenz gauge, show that at large distances , for which both and , the magnetic vector potential of the scattered radiation is
where with .
In the far-field, where , the electric and magnetic fields of the scattered radiation are given by
By calculating the Poynting vector of the scattered and incident radiation, show that the ratio of the time-averaged power scattered per unit solid angle to the time-averaged incident power per unit area (i.e. the differential cross-section) is
where and .
[You may assume that, in the Lorenz gauge, the retarded potential due to a localised current distribution is
where the retarded time
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Paper 2, Section II, B
2017 commentA cylinder of radius falls at speed without rotating through viscous fluid adjacent to a vertical plane wall, with its axis horizontal and parallel to the wall. The distance between the cylinder and the wall is . Use lubrication theory in a frame of reference moving with the cylinder to determine that the two-dimensional volume flux between the cylinder and the wall is
upwards, relative to the cylinder.
Determine an expression for the viscous shear stress on the cylinder. Use this to calculate the viscous force and hence the torque on the cylinder. If the cylinder is free to rotate, what does your result say about the sense of rotation of the cylinder?
[Hint: You may quote the following integrals:
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Paper 1, Section II, B
2017 commentFluid of density and dynamic viscosity occupies the region in Cartesian coordinates . A semi-infinite, dense array of cilia occupy the half plane , and apply a stress in the -direction on the adjacent fluid, working at a constant and uniform rate per unit area, which causes the fluid to move with steady velocity . Give a careful physical explanation of the boundary condition
paying particular attention to signs, where is the kinematic viscosity of the fluid. Why would you expect the fluid motion to be confined to a thin region near for sufficiently large values of ?
Write down the viscous-boundary-layer equations governing the thin region of fluid motion. Show that the flow can be approximated by a stream function
Determine the functions and . Show that the dimensionless function satisfies
What boundary conditions must be satisfied by ? By considering how the volume flux varies with downstream location , or otherwise, determine (with justification) the sign of the transverse flow .
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Paper 3, Section II, B
2017 commentA spherical bubble of radius a moves with velocity through a viscous fluid that is at rest far from the bubble. The pressure and velocity fields outside the bubble are given by
respectively, where is the dynamic viscosity of the fluid, is the position vector from the centre of the bubble and . Using suffix notation, or otherwise, show that these fields satisfy the Stokes equations.
Obtain an expression for the stress tensor for the fluid outside the bubble and show that the velocity field above also satisfies all the appropriate boundary conditions.
Compute the drag force on the bubble.
[Hint: You may use
where the integral is taken over the surface of a sphere of radius a and is the outward unit normal to the surface.]
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Paper 4, Section II, B
2017 commentA horizontal layer of inviscid fluid of density occupying flows with velocity above a horizontal layer of inviscid fluid of density occupying and flowing with velocity , in Cartesian coordinates . There are rigid boundaries at . The interface between the two layers is perturbed to position .
Write down the full set of equations and boundary conditions governing this flow. Derive the linearised boundary conditions appropriate in the limit . Solve the linearised equations to show that the perturbation to the interface grows exponentially in time if
Sketch the right-hand side of this inequality as a function of . Thereby deduce the minimum value of that makes the system unstable for all wavelengths.
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Paper 1, Section I, E
2017 commentCalculate the value of the integral
where stands for Principal Value and is a positive integer.
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Paper 2, Section I, E
2017 commentEuler's formula for the Gamma function is
Use Euler's formula to show
where is a constant.
Evaluate .
[Hint: You may use
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Paper 3, Section I, E
2017 commentFind all the singular points of the differential equation
and determine whether they are regular or irregular singular points.
By writing , find two linearly independent solutions to this equation.
Comment on the relationship of your solutions to the nature of the singular points of the original differential equation.
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Paper 4, Section I,
2017 commentConsider the differential equation
Laplace's method finds a solution of this differential equation by writing in the form
where is a closed contour.
Determine . Hence find two linearly independent real solutions of for real.
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Paper 2, Section II, E
2017 commentThe hypergeometric equation is represented by the Papperitz symbol
and has solution .
Functions and are defined by
and
where is not an integer.
Show that and obey the hypergeometric equation .
Explain why can be written in the form
where and are independent of but depend on and .
Suppose that
with and . Find expressions for and .
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Paper 1, Section II, E
2017 commentThe Riemann zeta function is defined by
for .
Show that
Let be defined by
where is the Hankel contour.
Show that provides an analytic continuation of for a range of which should be determined.
Hence evaluate .
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Paper 2, Section II, I
2017 comment(a) Define what it means for a finite field extension of a field to be separable. Show that is of the form for some .
(b) Let and be distinct prime numbers. Let . Express in the form and find the minimal polynomial of over .
(c) Give an example of a field extension of finite degree, where is not of the form . Justify your answer.
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Paper 3, Section II, I
2017 comment(a) Let be a finite field of characteristic . Show that is a finite Galois extension of the field of elements, and that the Galois group of over is cyclic.
(b) Find the Galois groups of the following polynomials:
(i) over .
(ii) over .
(iii) over .
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Paper 1, Section II, I
2017 comment(a) Let be a field and let . What does it mean for a field extension of to be a splitting field for over ?
Show that the splitting field for over is unique up to isomorphism.
(b) Find the Galois groups over the rationals for the following polynomials: (i) . (ii) .
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Paper 4, Section II, I
2017 comment(a) State the Fundamental Theorem of Galois Theory.
(b) What does it mean for an extension of to be cyclotomic? Show that a cyclotomic extension of is a Galois extension and prove that its Galois group is Abelian.
(c) What is the Galois group of over , where is a primitive 7 th root of unity? Identify the intermediate subfields , with , in terms of , and identify subgroups of to which they correspond. Justify your answers.
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Paper 2, Section II, D
2017 comment(a) The Friedmann-Robertson-Walker metric is given by
where and is the scale factor.
For , show that this metric can be written in the form
Calculate the equatorial circumference of the submanifold defined by constant and .
Calculate the proper volume, defined by , of the hypersurface defined by constant .
(b) The Friedmann equations are
where is the energy density, is the pressure, is the cosmological constant and dot denotes .
The Einstein static universe has vanishing pressure, . Determine and as a function of the density .
The Einstein static universe with and is perturbed by radiation such that
where and . Show that the Einstein static universe is unstable to this perturbation.
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Paper 1, Section II, D
2017 commentA static black hole in a five-dimensional spacetime is described by the metric
where is a constant.
A geodesic lies in the plane and has affine parameter . Show that
are both constants of motion. Write down a third constant of motion.
Show that timelike and null geodesics satisfy the equation
for some potential which you should determine.
Circular geodesics satisfy the equation . Calculate the values of for which circular null geodesics exist and for which circular timelike geodesics exist. Which are stable and which are unstable? Briefly describe how this compares to circular geodesics in the four-dimensional Schwarzschild geometry.
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Paper 3, Section II, D
2017 commentLet be a two-dimensional manifold with metric of signature .
(i) Let . Use normal coordinates at the point to show that one can choose two null vectors that form a basis of the vector space .
(ii) Consider the interval . Let be a null curve through and be the tangent vector to at . Show that the vector is either parallel to or parallel to .
(iii) Show that every null curve in is a null geodesic.
[Hint: You may wish to consider the acceleration .]
(iv) By providing an example, show that not every null curve in four-dimensional Minkowski spacetime is a null geodesic.
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Paper 4, Section II, D
2017 comment(a) In the transverse traceless gauge, a plane gravitational wave propagating in the direction is described by a perturbation of the Minkowski metric in Cartesian coordinates , where
and is a constant matrix. Spacetime indices in this question are raised or lowered with the Minkowski metric.
The energy-momentum tensor of a gravitational wave is defined to be
Show that and hence, or otherwise, show that energy and momentum are conserved.
(b) A point mass undergoes harmonic motion along the -axis with frequency and amplitude . Compute the energy flux emitted in gravitational radiation.
[Hint: The quadrupole formula for time-averaged energy flux radiated in gravitational waves is
\left\langle\frac{d E}{d t}\right\rangle=\frac{1}{5}\left\langle\dddot{Q}_{i j} \dddot{Q}_{i j}\right\rangle
where is the reduced quadrupole tensor.]
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Paper 3, Section II, H
2017 commentDefine the Ramsey numbers for integers . Show that exists for all . Show also that for all .
Let be fixed. Give a red-blue colouring of the edges of for which there is no red and no blue odd cycle. Show, however, that for any red-blue colouring of the edges of there must exist either a red or a blue odd cycle.
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Paper 2, Section II, H
2017 commentState and prove Hall's theorem about matchings in bipartite graphs.
Let be an matrix, with all entries non-negative reals, such that every row sum and every column sum is 1. By applying Hall's theorem, show that there is a permutation of such that for all .
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Paper 1, Section II, H
2017 commentLet be a graph of order satisfying . Show that is Hamiltonian.
Give an example of a planar graph , with , that is Hamiltonian, and also an example of a planar graph , with , that is not Hamiltonian.
Let be a planar graph with the property that the boundary of the unbounded face is a Hamilton cycle of . Prove that .
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Paper 4, Section II, H
2017 commentLet be a graph of maximum degree . Show the following:
(i) Every eigenvalue of satisfies .
(ii) If is regular then is an eigenvalue.
(iii) If is regular and connected then the multiplicity of as an eigenvalue is 1 .
(iv) If is regular and not connected then the multiplicity of as an eigenvalue is greater than 1 .
Let be the adjacency matrix of the Petersen graph. Explain why , where is the identity matrix and is the all-1 matrix. Find, with multiplicities, the eigenvalues of the Petersen graph.
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Paper 1, Section II, A
2017 commentDefine a Lie point symmetry of the first order ordinary differential equation 0. Describe such a Lie point symmetry in terms of the vector field that generates it.
Consider the -dimensional Hamiltonian system governed by the differential equation
Define the Poisson bracket . For smooth functions show that the associated Hamiltonian vector fields satisfy
If is a first integral of , show that the Hamiltonian vector field generates a Lie point symmetry of . Prove the converse is also true if has a fixed point, i.e. a solution of the form .
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Paper 2, Section II, A
2017 commentLet and be non-singular matrices depending on which are periodic in with period . Consider the associated linear problem
for the vector . On the assumption that these equations are compatible, derive the zero curvature equation for .
Let denote the matrix satisfying
where is the identity matrix. You should assume is unique. By considering , show that the matrix satisfies the Lax equation
Deduce that are first integrals.
By considering the matrices
show that the periodic Sine-Gordon equation has infinitely many first integrals. [You need not prove anything about independence.]
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Paper 3, Section II, A
2017 commentLet be a smooth solution to the equation
which decays rapidly as and let be the associated Schrödinger operator. You may assume and constitute a Lax pair for KdV.
Consider a solution to which has the asymptotic form
Find evolution equations for and . Deduce that is -independent.
By writing in the form
show that
Deduce that are first integrals of KdV.
By writing a differential equation for (with real), show that these first integrals are trivial when is even.
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Paper 3, Section II, F
2017 commentLet be a non-empty compact Hausdorff space and let be the space of real-valued continuous functions on .
(i) State the real version of the Stone-Weierstrass theorem.
(ii) Let be a closed subalgebra of . Prove that and implies that where the function is defined by . [You may use without proof that implies .]
(iii) Prove that is normal and state Urysohn's Lemma.
(iv) For any , define by for . Justifying your answer carefully, find
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Paper 2, Section II, F
2017 comment(a) Let be a normed vector space and a closed subspace with . Show that is nowhere dense in .
(b) State any version of the Baire Category theorem.
(c) Let be an infinite-dimensional Banach space. Show that cannot have a countable algebraic basis, i.e. there is no countable subset such that every can be written as a finite linear combination of elements of .
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Paper 1, Section II, F
2017 commentLet be a normed vector space over the real numbers.
(a) Define the dual space of and prove that is a Banach space. [You may use without proof that is a vector space.]
(b) The Hahn-Banach theorem states the following. Let be a real vector space, and let be sublinear, i.e., and for all and all . Let be a linear subspace, and let be linear and satisfy for all . Then there exists a linear functional such that for all and .
Using the Hahn-Banach theorem, prove that for any non-zero there exists such that and .
(c) Show that can be embedded isometrically into a Banach space, i.e. find a Banach space and a linear map with for all .
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Paper 4, Section II, F
2017 commentLet be a complex Hilbert space with inner product and let be a bounded linear map.
(i) Define the spectrum , the point spectrum , the continuous spectrum , and the residual spectrum .
(ii) Show that is self-adjoint and that . Show that if is compact then so is .
(iii) Assume that is compact. Prove that has a singular value decomposition: for or , there exist orthonormal systems and and such that, for any ,
[You may use the spectral theorem for compact self-adjoint linear operators.]
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Paper 3, Section II, H
2017 commentState and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Indicate clearly where in your proof you have made use of the Axiom of Choice.
Show that has a basis as a vector space over .
Let be a vector space over . Show that all bases of have the same cardinality.
[Hint: How does the cardinality of relate to the cardinality of a given basis?]
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Paper 2, Section II, H
2017 commentGive the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent.
Which of the following are always true for ordinals and and which can be false? Give proofs or counterexamples as appropriate.
(i)
(ii)
(iii)
(iv) If then
(v) If then
[In parts (iv) and (v) you may assume without proof that ordinal multiplication is associative.]
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Paper 4, Section II, H
2017 commentProve that every set has a transitive closure. [If you apply the Axiom of Replacement to a function-class , you must explain clearly why is indeed a function-class.]
State the Axiom of Foundation and the Principle of -Induction, and show that they are equivalent (in the presence of the other axioms of ).
State the -Recursion Theorem.
Sets are defined for each ordinal by recursion, as follows: is the set of all countable subsets of , and for a non-zero limit. Does there exist an with ? Justify your answer.
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Paper 1, Section II, H
2017 commentState the Completeness Theorem for Propositional Logic.
[You do not need to give definitions of the various terms involved.]
State the Compactness Theorem and the Decidability Theorem, and deduce them from the Completeness Theorem.
A set of propositions is called finitary if there exists a finite set of propositions such that . Give examples to show that an infinite set of propositions may or may not be finitary.
Now let and be sets of propositions such that every valuation is a model of exactly one of and . Show that there exist finite subsets of and of with , and deduce that and are finitary.
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Paper 1, Section I, B
2017 commentA model of insect dispersal and growth in one spatial dimension is given by
where and are constants, , and may be positive or negative.
By setting , where is some time-like variable satisfying , show that a suitable choice of yields
where subscript denotes differentiation with respect to or .
Consider a similarity solution of the form where . Show that must satisfy
[You may use the fact that these are solved by
where
For , what is the maximum distance from the origin that insects ever reach? Give your answer in terms of and .
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Paper 2, Section I, B
2017 commentA bacterial nutrient uptake model is represented by the reaction system
where the are rate constants. Let and represent the concentrations of and respectively. Initially and . Write down the governing differential equation system for the concentrations.
Either by using the differential equations or directly from the reaction system above, find two invariant quantities. Use these to simplify the system to
By setting and and rescaling time, show that the system can be written as
where and and should be given. Give the initial conditions for and .
[Hint: Note that is equivalent to in reaction systems.]
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Paper 3, Section I, B
2017 commentA stochastic birth-death process has a master equation given by
where is the probability that there are individuals in the population at time for and for .
Give the corresponding Fokker-Planck equation for this system.
Use this Fokker-Planck equation to find expressions for and .
[Hint: The general form for a Fokker-Planck equation in is
You may use this general form, stating how and are constructed. Alternatively, you may derive a Fokker-Plank equation directly by working from the master equation.]
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Paper 4, Section I, B
2017 commentConsider an epidemic model with host demographics (natural births and deaths).
The system is given by
where are the susceptibles, are the infecteds, is the total population size and the parameters and are positive. The basic reproduction ratio is defined as
Show that the system has an endemic equilibrium (where the disease is present) for . Show that the endemic equilibrium is stable.
Interpret the meaning of the case and show that in this case the approximate period of (decaying) oscillation around the endemic equilibrium is given by
Suppose now a vaccine is introduced which is given to some proportion of the population at birth, but not enough to eradicate the disease. What will be the effect on the period of (decaying) oscillations?
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Paper 3, Section II, B
2017 commentIn a discrete-time model, adults and larvae of a population at time are represented by and respectively. The model is represented by the equations
You may assume that and . Give an explanation of what each of the terms represents, and hence give a description of the population model.
By combining the equations to describe the dynamics purely in terms of the adults, find all equilibria of the system. Show that the equilibrium with the population absent is unstable exactly when there exists an equilibrium with the population present .
Give the condition on and for the equilibrium with to be stable, and sketch the corresponding region in the plane.
What happens to the population close to the boundaries of this region?
If this model was modified to include stochastic effects, briefly describe qualitatively the likely dynamics near the boundaries of the region found above.
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Paper 4, Section II, B
2017 commentAn activator-inhibitor system is described by the equations
where .
Find and sketch the range of for which the spatially homogeneous system has a stable stationary solution with and .
Considering spatial perturbations of the form about the solution found above, find conditions for the system to be unstable. Sketch this region in the -plane for fixed (for a value of such that the region is non-empty).
Show that , the critical wavenumber at the onset of the instability, is given by
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Paper 2, Section II, 18H
2017 comment(a) Let be a number field, the ring of integers in the units in the number of real embeddings of , and the number of pairs of complex embeddings of .
Define a group homomorphism with finite kernel, and prove that the image is a discrete subgroup of .
(b) Let where is a square-free integer. What is the structure of the group of units of ? Show that if is divisible by a prime then every unit of has norm . Find an example of with a unit of norm .
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Paper 1, Section II, H
2017 commentLet be the ring of integers in a number field , and let be a non-zero ideal of .
(a) Show that .
(b) Show that is a finite abelian group.
(c) Show that if has , then .
(d) Suppose , and , with and . Show that is principal.
[You may assume that has an integral basis.]
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Paper 4, Section II, H
2017 comment(a) Write down , when , and or . [You need not prove your answer.]
Let , where is a square-free integer. Find an integral basis of [Hint: Begin by considering the relative traces , for a quadratic subfield of
(b) Compute the ideal class group of .
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Paper 1, Section , G
2017 commentDefine the Legendre symbol .
State Gauss' lemma and use it to compute where is an odd prime.
Show that if is a power of 2 , and is a prime dividing , then .
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Paper 3, Section I, G
2017 commentExplain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler pseudoprime to the base if and only if . How many such bases are there? Show that the bases for which 65 is a strong pseudoprime do not form a subgroup of .
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Paper 4, Section I, G
2017 commentShow that, for a real number,
Hence prove that
where is a constant you should make explicit.
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Paper 2, Section I, G
2017 commentState and prove Legendre's formula for . Use it to compute .
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Paper 3, Section II, G
2017 commentLet be a positive integer which is not a square. Assume that the continued fraction expansion of takes the form .
(a) Define the convergents , and show that and are coprime.
(b) The complete quotients may be written in the form , where and are rational numbers. Use the relation
to find formulae for and in terms of the 's and 's. Deduce that and are integers.
(c) Prove that Pell's equation has infinitely many solutions in integers and .
(d) Find integers and satisfying .
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Paper 4, Section II, 10G
2017 comment(a) State Dirichlet's theorem on primes in arithmetic progression.
(b) Let be the discriminant of a binary quadratic form, and let be an odd prime. Show that is represented by some binary quadratic form of discriminant if and only if is soluble.
(c) Let and . Show that and each represent infinitely many primes. Are there any primes represented by both and ?
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Paper 2, Section II, A
2017 commentThe Poisson equation in the unit square , equipped with the zero Dirichlet boundary conditions on , is discretized with the nine-point formula:
where , and are the grid points with .
(i) Find the order of the local truncation error of the approximation.
(ii) Prove that the order of the truncation error is smaller if satisfies the Laplace equation .
(iii) Show that the modified nine-point scheme
has a truncation error of the same order as in part (ii).
(iv) Let be a solution to the system of linear equations arising from the modified nine-point scheme in part (iii). Further, let be the exact solution and let be the error of approximation at grid points. Prove that there exists a constant such that
[Hint: The nine-point discretization of can be written as
where is the five-point discretization and
[Hint: The matrix A of the nine-point scheme is symmetric, with the eigenvalues
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Paper 1, Section II, A
2017 commentState the Householder-John theorem and explain how it can be used in designing iterative methods for solving a system of linear equations . [You may quote other relevant theorems if needed.]
Consider the following iterative scheme for solving . Let , where is the diagonal part of , and and are the strictly lower and upper triangular parts of , respectively. Then, with some starting vector , the scheme is as follows:
Prove that if is a symmetric positive definite matrix and , then, for any , the above iteration converges to the solution of the system .
Which method corresponds to the case
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Paper 3, Section II, A
2017 commentLet be a real symmetric matrix with real and distinct eigenvalues and a corresponding orthogonal basis of normalized real eigenvectors .
To estimate the eigenvector of whose eigenvalue is , the power method with shifts is employed which has the following form:
Three versions of this method are considered:
(i) no shift: ;
(ii) single shift: ;
(iii) double shift: .
Assume that , where is very small, so that the terms are negligible, and that contains substantial components of all the eigenvectors.
By considering the approximation after iterations in the form
find as a function of for each of the three versions of the method.
Compare the convergence rates of the three versions of the method, with reference to the number of iterations needed to achieve a prescribed accuracy.
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Paper 4, Section II, A
2017 comment(a) The diffusion equation
is approximated by the Crank-Nicolson scheme
with . Here , and is an approximation to . Assuming that , show that the above scheme can be written in the form
where and the real matrices and should be found. Using matrix analysis, find the range of for which the scheme is stable.
[Hint: All Toeplitz symmetric tridiagonal (TST) matrices have the same set of orthogonal eigenvectors, and a TST matrix with the elements and has the eigenvalues . ]
(b) The wave equation
with given initial conditions for and , is approximated by the scheme
with the Courant number now . Applying the Fourier technique, find the range of for which the method is stable.
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Paper 2, Section II, K
2017 commentDuring each of time periods a venture capitalist, Vicky, is presented with an investment opportunity for which the rate of return for that period is a random variable; the rates of return in successive periods are independent identically distributed random variables with distributions concentrated on . Thus, if is Vicky's capital at period , then , where is the proportion of her capital she chooses to invest at period , and is the rate of return for period . Vicky desires to maximize her expected yield over periods, where the yield is defined as , and and are respectively her initial and final capital.
(a) Express the problem of finding an optimal policy in a dynamic programming framework.
(b) Show that in each time period, the optimal strategy can be expressed in terms of the quantity which solves the optimization problem . Show that if . [Do not calculate explicitly.]
(c) Compare her optimal policy with the policy which maximizes her expected final capital .
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Paper 3, Section II, K
2017 commentA particle follows a discrete-time trajectory on given by
for . Here is a fixed integer, is a real constant, and are the position of the particle and control action at time , respectively, and is a sequence of independent random vectors with
Find the optimal control, i.e. the control action , defined as a function of , that minimizes
where is given.
On which of and does the optimal control depend?
Find the limiting form of the optimal control as , and the minimal average cost per unit time.
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Paper 4, Section II,
2017 commentA file of gigabytes (GB) is to be transmitted over a communications link. At each time the sender can choose a transmission rate within the range GB per second. The charge for transmitting at rate at time is . The function is fully known at time . If it takes a total time to transmit the file then there is a delay cost of . Thus and are to be chosen to minimize
where and . Using Pontryagin's maximum principle, or otherwise, show that a property of the optimal policy is that there exists such that if and if .
Show that the optimal and are related by .
Suppose and . Show that it is optimal to transmit at a constant rate between times , where is the unique positive solution to the equation
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Paper 1, Section II, C
2017 commentThe position and momentum operators of the harmonic oscillator can be written as
where is the mass, is the frequency and the Hamiltonian is
Assuming that
derive the commutation relations for and . Construct the Hamiltonian in terms of and . Assuming that there is a unique ground state, explain how all other energy eigenstates can be constructed from it. Determine the energy of each of these eigenstates.
Consider the modified Hamiltonian
where is a dimensionless parameter. Use perturbation theory to calculate the modified energy levels to second order in , quoting any standard formulae that you require. Show that the modified Hamiltonian can be written as
Assuming , calculate the modified energies exactly. Show that the results are compatible with those obtained from perturbation theory.
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Paper 2, Section II, C
2017 commentLet be a set of Hermitian operators obeying
where is any unit vector. Show that implies that
for any vectors a and . Explain, with reference to the properties , how can be related to the intrinsic angular momentum for a particle of spin .
Show that the operators are Hermitian and obey
Show how can be used to write any state as a linear combination of eigenstates of . Use this to deduce that if the system is in a normalised state when is measured, then the results will be obtained with probabilities
If is a state corresponding to the system having spin up along a direction defined by a unit vector , show that a measurement will find the system to have spin up along with probability .
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Paper 3, Section II, C
2017 commentThe angular momentum operators obey the commutation relations
where .
A quantum mechanical system involves the operators and such that
Define and . Show that and obey the same commutation relations as and .
Suppose that the system is in the state such that . Show that is an eigenstate of . Let . Show that is an eigenstate of and find the eigenvalue. How many other states do you expect to find with same value of ? Find them.
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Paper 4, Section II, C
2017 commentThe Hamiltonian for a quantum system in the Schrödinger picture is
where is independent of time and the parameter is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.
Let and be eigenstates of with distinct eigenvalues and respectively. Show that if the system was in the state in the remote past, then the probability of measuring it to be in a different state at a time is
Let the system be a simple harmonic oscillator with , where . Let be the ground state which obeys . Suppose
with . In the remote past the system was in the ground state. Find the probability, to lowest non-trivial order in , for the system to be in the first excited state in the far future.
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Paper 2, Section II,
2017 commentWe consider the problem of estimating in the model , where
Here is the indicator of the set , and is known. This estimation is based on a sample of i.i.d. , and we denote by the ordered sample.
(a) Compute the mean and the variance of . Construct an unbiased estimator of taking the form , where , specifying .
(b) Show that is consistent and find the limit in distribution of . Justify your answer, citing theorems that you use.
(c) Find the maximum likelihood estimator of . Compute for all real . Is unbiased?
(d) For , show that has a limit in for some . Give explicitly the value of and the limit. Why should one favour using over ?
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Paper 3, Section II,
2017 commentWe consider the problem of estimating an unknown in a statistical model where , based on i.i.d. observations whose distribution has p.d.f. .
In all the parts below you may assume that the model satisfies necessary regularity conditions.
(a) Define the score function of . Prove that has mean 0 .
(b) Define the Fisher Information . Show that it can also be expressed as
(c) Define the maximum likelihood estimator of . Give without proof the limits of and of ) (in a manner which you should specify). [Be as precise as possible when describing a distribution.]
(d) Let be a continuously differentiable function, and another estimator of such that with probability 1 . Give the limits of and of (in a manner which you should specify).
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Paper 4, Section II,
2017 commentFor the statistical model , where is a known, positive-definite matrix, we want to estimate based on i.i.d. observations with distribution .
(a) Derive the maximum likelihood estimator of . What is the distribution of ?
(b) For , construct a confidence region such that .
(c) For , compute the maximum likelihood estimator of for the following parameter spaces:
(i) .
(ii) for some unit vector .
(d) For , we want to test the null hypothesis (i.e. against the composite alternative . Compute the likelihood ratio statistic and give its distribution under the null hypothesis. Compare this result with the statement of Wilks' theorem.
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Paper 1, Section II,
2017 commentFor a positive integer , we want to estimate the parameter in the binomial statistical model , based on an observation .
(a) Compute the maximum likelihood estimator for . Show that the posterior distribution for under a uniform prior on is , and specify and . [The p.d.f. of is given by
(b) (i) For a risk function , define the risk of an estimator of , and the Bayes risk under a prior for .
(ii) Under the loss function
find a Bayes optimal estimator for the uniform prior. Give its risk as a function of .
(iii) Give a minimax optimal estimator for the loss function given above. Justify your answer.
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Paper 2, Section II, J
2017 comment(a) Give the definition of the Fourier transform of a function .
(b) Explain what it means for Fourier inversion to hold.
(c) Prove that Fourier inversion holds for . Show all of the steps in your computation. Deduce that Fourier inversion holds for Gaussian convolutions, i.e. any function of the form where and .
(d) Prove that any function for which Fourier inversion holds has a bounded, continuous version. In other words, there exists bounded and continuous such that for a.e. .
(e) Does Fourier inversion hold for ?
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Paper 3, Section II, J
2017 comment(a) Suppose that is a sequence of random variables on a probability space . Give the definition of what it means for to be uniformly integrable.
(b) State and prove Hölder's inequality.
(c) Explain what it means for a family of random variables to be bounded. Prove that an bounded sequence is uniformly integrable provided .
(d) Prove or disprove: every sequence which is bounded is uniformly integrable.
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Paper 4, Section II, J
2017 comment(a) Suppose that is a finite measure space and is a measurable map. Prove that defines a measure on .
(b) Suppose that is a -system which generates . Using Dynkin's lemma, prove that is measure-preserving if and only if for all .
(c) State Birkhoff's ergodic theorem and the maximal ergodic lemma.
(d) Consider the case where is Lebesgue measure on . Let be the following map. If is the binary expansion of (where we disallow infinite sequences of ), then where and are respectively the even and odd elements of .
(i) Prove that is measure-preserving. [You may assume that is measurable.]
(ii) Prove or disprove: is ergodic.
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Paper 1, Section II, J
2017 comment(a) Give the definition of the Borel -algebra on and a Borel function where is a measurable space.
(b) Suppose that is a sequence of Borel functions which converges pointwise to a function . Prove that is a Borel function.
(c) Let be the function which gives the th binary digit of a number in ) (where we do not allow for the possibility of an infinite sequence of 1 s). Prove that is a Borel function.
(d) Let be the function such that for is equal to the number of digits in the binary expansions of which disagree. Prove that is non-negative measurable.
(e) Compute the Lebesgue measure of , i.e. the set of pairs of numbers in whose binary expansions disagree in a finite number of digits.
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Paper 2, Section II, G
2017 commentIn this question you may assume the following result. Let be a character of a finite group and let . If is a rational number, then is an integer.
(a) If and are positive integers, we denote their highest common factor by . Let be an element of order in the finite group . Suppose that is conjugate to for all with and . Prove that is an integer for all characters of .
[You may use the following result without proof. Let be an th root of unity. Then
is an integer.]
Deduce that all the character values of symmetric groups are integers.
(b) Let be a group of odd order.
Let be an irreducible character of with . Prove that
where is an algebraic integer. Deduce that .
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Paper 3, Section II, G
2017 comment(a) State Burnside's theorem.
(b) Let be a non-trivial group of prime power order. Show that if is a non-trivial normal subgroup of , then .
Deduce that a non-abelian simple group cannot have an abelian subgroup of prime power index.
(c) Let be a representation of the finite group over . Show that is a linear character of . Assume that for some . Show that has a normal subgroup of index 2 .
Now let be a group of order , where is an odd integer. By considering the regular representation of , or otherwise, show that has a normal subgroup of index
Deduce that if is a non-abelian simple group of order less than 80 , then has order 60 .
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Paper 1, Section II, G
2017 comment(a) Prove that if there exists a faithful irreducible complex representation of a finite group , then the centre is cyclic.
(b) Define the permutations by
and let .
(i) Using the relations and , prove that has order 18 .
(ii) Suppose that and are complex cube roots of unity. Prove that there is a (matrix) representation of over such that
(iii) For which values of is faithful? For which values of is irreducible?
(c) Note that is a normal subgroup of which is isomorphic to . By inducing linear characters of this subgroup, or otherwise, obtain the character table of .
Deduce that has the property that is cyclic but has no faithful irreducible representation over .
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Paper 4, Section II, G
2017 commentLet and let be the vector space of complex homogeneous polynomials of degree in two variables.
(a) Prove that has the structure of an irreducible representation for .
(b) State and prove the Clebsch-Gordan theorem.
(c) Quoting without proof any properties of symmetric and exterior powers which you need, decompose and into irreducible -spaces.
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Paper 2, Section II, F
2017 commentLet be a non-constant elliptic function with respect to a lattice . Let be a fundamental parallelogram whose boundary contains no zeros or poles of . Show that the number of zeros of in is the same as the number of poles of in , both counted with multiplicities.
Suppose additionally that is even. Show that there exists a rational function such that , where is the Weierstrass -function.
Suppose is a non-constant elliptic function with respect to a lattice , and is a meromorphic antiderivative of , so that . Is it necessarily true that is an elliptic function? Justify your answer.
[You may use standard properties of the Weierstrass -function throughout.]
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Paper 3, Section II, F
2017 commentLet be a positive even integer. Consider the subspace of given by the equation , where are coordinates in , and let be the restriction of the projection map to the first factor. Show that has the structure of a Riemann surface in such a way that becomes an analytic map. If denotes projection onto the second factor, show that is also analytic. [You may assume that is connected.]
Find the ramification points and the branch points of both and . Compute the ramification indices at the ramification points.
Assume that, by adding finitely many points, it is possible to compactify to a Riemann surface such that extends to an analytic map . Find the genus of (as a function of ).
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Paper 1, Section II, F
2017 commentBy considering the singularity at , show that any injective analytic map has the form for some and .
State the Riemann-Hurwitz formula for a non-constant analytic map of compact Riemann surfaces and , explaining each term that appears.
Suppose is analytic of degree 2. Show that there exist Möbius transformations and such that
is the map given by .
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Paper 1, Section I, J
2017 commentThe dataset ChickWeights records the weight of a group of chickens fed four different diets at a range of time points. We perform the following regressions in .

(i) Which hypothesis test does the following command perform? State the degrees of freedom, and the conclusion of the test.

(ii) Define a diagnostic plot that might suggest the logarithmic transformation of the response in fit2.
(iii) Define the dashed line in the following plot, generated with the command plot(fit3). What does it tell us about the data point 579 ?

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Paper 2, Section I, J
2017 commentA statistician is interested in the power of a -test with level in linear regression; that is, the probability of rejecting the null hypothesis with this test under an alternative with .
(a) State the distribution of the least-squares estimator , and hence state the form of the -test statistic used.
(b) Prove that the power does not depend on the other coefficients for .
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Paper 3, Section I, J
2017 commentFor Fisher's method of Iteratively Reweighted Least-Squares and Newton-Raphson optimisation of the log-likelihood, the vector of parameters is updated using an iteration
for a specific function . How is defined in each method?
Prove that they are identical in a Generalised Linear Model with the canonical link function.
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Paper 4, Section I, J
2017 commentA Cambridge scientist is testing approaches to slow the spread of a species of moth in certain trees. Two groups of 30 trees were treated with different organic pesticides, and a third group of 30 trees was kept under control conditions. At the end of the summer the trees are classified according to the level of leaf damage, obtaining the following contingency table.

Which of the following Generalised Linear Model fitting commands is appropriate for these data? Why? Describe the model being fit.

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Paper 1, Section II, J
2017 commentThe Cambridge Lawn Tennis Club organises a tournament in which every match consists of 11 games, all of which are played. The player who wins 6 or more games is declared the winner.
For players and , let be the total number of games they play against each other, and let be the number of these games won by player . Let and be the corresponding number of matches.
A statistician analysed the tournament data using a Binomial Generalised Linear Model (GLM) with outcome . The probability that wins a game against is modelled by
with an appropriate corner point constraint. You are asked to re-analyse the data, but the game-level results have been lost and you only know which player won each match.
We define a new GLM for the outcomes with and , where the are defined in . That is, is the log-odds that wins a game against , not a match.
Derive the form of the new link function . [You may express your answer in terms of a cumulative distribution function.]
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Paper 4, Section II, J
2017 commentThe dataset diesel records the number of diesel cars which go through a block of Hills Road in 6 disjoint periods of 30 minutes, between 8AM and 11AM. The measurements are repeated each day for 10 days. Answer the following questions based on the code below, which is shown with partial output.
(a) Can we reject the model fit. 1 at a level? Justify your answer.
(b) What is the difference between the deviance of the models fit. 2 and fit.3?
(c) Which of fit. 2 and fit. 3 would you use to perform variable selection by backward stepwise selection? Why?
(d) How does the final plot differ from what you expect under the model in fit.2? Provide a possible explanation and suggest a better model.
head (diesel)
period num.cars day
fit. glm(num.cars period, data=diesel, family=poisson)
summary (fit.1)
Deviance Residuals:
Min 1Q Median 3Q Max
Coefficients:
Estimate Std. Error value
(Intercept)
period
Signif. codes: 0 ? ? ? ? ?.? ? ? 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: on 59 degrees of freedom
Residual deviance: on 58 degrees of freedom
AIC:
diesel$period.factor = factor(diesel$period)
fit. glm (num.cars period.factor, data=diesel, family=poisson)
(fit.2)
Coefficients:
Estimate Std. Error z value

Part II, List of Questions
[TURN OVER
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Paper 4, Section II, D
2017 commentThe van der Waals equation of state is
where is the pressure, is the volume divided by the number of particles, is the temperature, is Boltzmann's constant and are positive constants.
(i) Prove that the Gibbs free energy satisfies . Hence obtain an expression for and use it to explain the Maxwell construction for determining the pressure at which the gas and liquid phases can coexist at a given temperature.
(ii) Explain what is meant by the critical point and determine the values corresponding to this point.
(iii) By defining and , derive the law of corresponding states:
(iv) To investigate the behaviour near the critical point, let and , where and are small. Expand to cubic order in and hence show that
At fixed small , let and be the values of corresponding to the liquid and gas phases on the co-existence curve. By changing the integration variable from to , use the Maxwell construction to show that . Deduce that, as the critical point is approached along the co-existence curve,
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Paper 1, Section II, D
2017 commentExplain what is meant by the microcanonical ensemble for a quantum system. Sketch how to derive the probability distribution for the canonical ensemble from the microcanonical ensemble. Under what physical conditions should each type of ensemble be used?
A paramagnetic solid contains atoms with magnetic moment , where is a positive constant and is the intrinsic angular momentum of the atom. In an applied magnetic field , the energy of an atom is . Consider . Each atom has total angular momentum , so the possible values of are .
Show that the partition function for a single atom is
where .
Compute the average magnetic moment of the atom. Sketch for , and on the same graph.
The total magnetization is , where is the number of atoms. The magnetic susceptibility is defined by
Show that the solid obeys Curie's law at high temperatures. Compute the susceptibility at low temperatures and give a physical explanation for the result.
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Paper 2, Section II, 34D
2017 comment(a) The entropy of a thermodynamic ensemble is defined by the formula
where is the Boltzmann constant. Explain what is meant by in this formula. Write down an expression for in the grand canonical ensemble, defining any variables you need. Hence show that the entropy is related to the grand canonical partition function by
(b) Consider a gas of non-interacting fermions with single-particle energy levels .
(i) Show that the grand canonical partition function is given by
(ii) Assume that the energy levels are continuous with density of states , where and are positive constants. Prove that
and give expressions for the constant and the function .
(iii) The gas is isolated and undergoes a reversible adiabatic change. By considering the ratio , prove that remains constant. Deduce that and remain constant in this process, where and are constants whose values you should determine.
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Paper 3, Section II, D
2017 comment(a) Describe the Carnot cycle using plots in the -plane and the -plane. In which steps of the cycle is heat absorbed or emitted by the gas? In which steps is work done on, or by, the gas?
(b) An ideal monatomic gas undergoes a reversible cycle described by a triangle in the -plane with vertices at the points with coordinates and respectively. The cycle is traversed in the order .
(i) Write down the equation of state and an expression for the internal energy of the gas.
(ii) Derive an expression relating to and . Use your expression to calculate the heat supplied to, or emitted by, the gas along and .
(iii) Show that heat is supplied to the gas along part of the line , and is emitted by the gas along the other part of the line.
(iv) Calculate the efficiency where is the total work done by the cycle and is the total heat supplied.
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Paper 2, Section II,
2017 comment(a) What is a Brownian motion?
(b) Let be a Brownian motion. Show that the process , , is also a Brownian motion.
(c) Let . Show that for all (i.e. and have the same laws). Conclude that a.s.
(d) Show that .
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Paper 3, Section II, J
2017 comment(a) State the fundamental theorem of asset pricing for a multi-period model.
Consider a market model in which there is no arbitrage, the prices for all European put and call options are already known and there is a riskless asset with for some . The holder of a so-called 'chooser option' has the right to choose at a preassigned time between a European call and a European put option on the same asset , both with the same strike price and the same maturity . [We assume that at time the holder will take the option having the higher price at that time.]
(b) Show that the payoff function of the chooser option is given by
(c) Show that the price of the chooser option is given by
where and denote the price of a European call and put option, respectively, with strike and maturity .
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Paper 4, Section II, J
2017 comment(a) Describe the (Cox-Ross-Rubinstein) binomial model. When is the model arbitragefree? How is the equivalent martingale measure characterised in this case?
(b) What is the price and the hedging strategy for any given contingent claim in the binomial model?
(c) For any fixed and , the payoff function of a forward-start-option is given by
Find a formula for the price of the forward-start-option in the binomial model.
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Paper 1, Section II, J
2017 comment(a) What does it mean to say that is a martingale?
(b) Let be independent random variables on with and . Further, let
where
Show that is a martingale with respect to the natural filtration .
(c) State and prove the optional stopping theorem for a bounded stopping time .
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Paper 2, Section I, F
2017 commentAre the following statements true or false? Give reasons, quoting any theorems that you need.
(i) There is a sequence of polynomials with uniformly on as .
(ii) If is continuous, then there is a sequence of polynomials with for each as .
(iii) If is continuous with as , then there is a sequence of polynomials with uniformly on as .
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Paper 4, Section I, F
2017 commentIf , set
where is an integer and . Let .
If is also irrational, write down the continued fraction expansion in terms of where .
Let be a random variable taking values in with probability density function
Show that has the same distribution as .
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Paper 1, Section I,
2017 commentState Liouville's theorem on the approximation of algebraic numbers by rationals.
Suppose that we have a sequence with . State and prove a necessary and sufficient condition on the for
to be transcendental.
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Paper 3, Section I,
2017 comment(a) Suppose that is a continuous function such that there exists a with for all . By constructing a suitable map from the closed unit disc into itself, show that there exists a with .
(b) Show that is surjective.
(c) Show that the result of part (b) may be false if we drop the condition that is continuous.
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Paper 2, Section II, F
2017 commentState and prove Baire's category theorem for complete metric spaces. Give an example to show that it may fail if the metric space is not complete.
Let be a sequence of continuous functions such that converges for all . Show that if is fixed we can find an and a non-empty open interval such that for all and all .
Let be defined by
Show that we cannot find continuous functions with for each as
Define a sequence of continuous functions and a discontinuous function with for each as .
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Paper 4, Section II,
2017 comment(a) Suppose that is continuous with and for all . Show that if (with real) we can define a continuous function such that and . Hence define the winding number of around 0 .
(b) Show that can take any integer value.
(c) If and satisfy the requirements of the definition, and , show that
(d) If and satisfy the requirements of the definition and for all , show that
(e) State and prove a theorem that says that winding number is unchanged under an appropriate homotopy.
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Paper 2, Section II, B
2017 commentShow that, for a one-dimensional flow of a perfect gas (with ) at constant entropy, the Riemann invariants are constant along characteristics
Define a simple wave. Show that in a right-propagating simple wave
In some circumstances, dissipative effects may be modelled by
where is a positive constant. Suppose also that is prescribed at for all , say . Demonstrate that, unless a shock develops, a solution of the form
can be found, where, for each and is determined implicitly as the solution of the equation
Deduce that, despite the presence of dissipative effects, a shock will still form at some unless , where
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Paper 1, Section II, B
2017 commentDerive the wave equation governing the pressure disturbance , for linearised, constant entropy sound waves in a compressible inviscid fluid of density and sound speed , which is otherwise at rest.
Consider a harmonic acoustic plane wave with wavevector and unit-amplitude pressure disturbance. Determine the resulting velocity field .
Consider such an acoustic wave incident from on a thin elastic plate at . The regions and are occupied by gases with densities and , respectively, and sound speeds and , respectively. The kinematic boundary conditions at the plate are those appropriate for an inviscid fluid, and the (linearised) dynamic boundary condition is
where and are the mass and bending moment per unit area of the plate, and (with ) is its perturbed position. Find the amplitudes of the reflected and transmitted pressure perturbations, expressing your answers in terms of the dimensionless parameter
(i) If and , under what condition is the incident wave perfectly transmitted?
(ii) If , comment on the reflection coefficient, and show that waves incident at a sufficiently large angle are reflected as if from a pressure-release surface (i.e. an interface where ), no matter how large the plate mass and bending moment may be.
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Paper 3, Section II, B
2017 commentWaves propagating in a slowly-varying medium satisfy the local dispersion relation in the standard notation. Derive the ray-tracing equations
governing the evolution of a wave packet specified by , where . A formal justification is not required, but the meaning of the notation should be carefully explained.
The dispersion relation for two-dimensional, small amplitude, internal waves of wavenumber , relative to Cartesian coordinates with vertical, propagating in an inviscid, incompressible, stratified fluid that would otherwise be at rest, is given by
where is the Brunt-Väisälä frequency and where you may assume that and . Derive the modified dispersion relation if the fluid is not at rest, and instead has a slowly-varying mean flow .
In the case that and is constant, show that a disturbance with wavenumber generated at will propagate upwards but cannot go higher than a critical level , where is equal to the apparent wave speed in the -direction. Find expressions for the vertical wave number as from below, and show that it takes an infinite time for the wave to reach the critical level.
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Paper 4, Section II, 38B
2017 commentConsider the Rossby-wave equation
where and are real constants. Find and sketch the dispersion relation for waves with wavenumber and frequency . Find and sketch the phase velocity and the group velocity , and identify in which direction(s) the wave crests travel, and the corresponding direction(s) of the group velocity.
Write down the solution with initial value
where is real and . Use the method of stationary phase to obtain leading-order approximations to for large , with having the constant value , for
(i) ,
(ii) ,
where the solutions for the stationary points should be left in implicit form. [It is helpful to note that .]
Briefly discuss the nature of the solution for and . [Detailed calculations are not required.]
[Hint: You may assume that
for